Two-Sided Ideals and Congruences in the Ring of Bounded Operators

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					Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert

         J. W. Calkin

         The Annals of Mathematics, 2nd Ser., Vol. 42, No. 4. (Oct., 1941), pp. 839-873.

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                                                                                                       Sat Feb 9 13:21:27 2008
Vol. 42 No. 4, October, 1941



                                (Received August 29, 1940)

   The developments of the present paper center around the observation that
the ring 93 of bounded everywhere defined operators in Hilbert space contains
non-trivial two-sided ideals.' This fact, which has escaped all but oblique notice
in the develcpment of the theory of operators, is of course fundamental from the
point of view of algebra and a t the same t>imedifferentiates $8 sharply from the
ring of all linear operators over a unitary space with finite dimension number.
   As examples of two-sided ideals in $8 we may mention here the class of all
operators -4 such that % ( A ) , the range of A , has a finite dimension number, the
ciass of all operators of Hilbert-Schmidt type,2 and the class 3 of all totally
continuous operators. Except for the ideal (0), every two-sided ideal in 93
contains the first ideal mentioned, and except for the ideal '3 itself, every two-
sided ideal in % is contained in the ideal X hloreover, on the basis of the special
spectral properties of the self-adjoint members of '3; it is possible to characterize
every t&o-sided ideal in 93 very simply in terms of the spectra of its nonnegative
self-adjoint elements; for both the formulation and the proof of this result, which
together with the facts mentioned above is discussed in $1, the author is indebted
to J . v. Keumann.
   The restriction of our attention to those ideals in 93 which are two-sided is
basic for the points which we wish to develop ; the two-sidedness compensates for
the absence of commutativity in 93 in such a way as to permit the construction
of quotient rings by the standard methods of abstract algebra.3 These rings,
which are of course homomorphs of 93 with respect to addition and multiplica-
tion, are also homomorphs of 9 with respect to the operation *, and exhibit all
of the formal properties of matrix algebras. This is established in 82, and there
also various properties of the associated congruences in 93 are discussed.
   The remainder of the paper deals solely with the quotient ring 93/'Y, \\\.here9-
is the ideal of totally continuous operators, and the associated congruence in '3.
For essentially topological reasons, this is the only one of t>hequotient rings in

    An additive subset 9of 9 is a left (right) ideal if it contains A B ( B A ) for all A in g
and B in 3. If 9 is both a left and right ideal, it is called two-sided. See references [I]
and [19]a t the end of the paper for the elementary properties of ideals.
    For a discussion in abstract terms of operators of this type (operators of "finite norm"),
see reference [17], pp. 65-70.
     See, for example, [I], pp. 252-253, [19], pp. 5&57.
question which a t present appears susceptible of deep analysis. For while
there is in general no apparent way of introducing a topology in quotient rings
over 58, the ring 58/Tis actually a complete metric space, the norm deriving
very simply from the norm (i.e., bound) in 9 itself (53).
   Moreover, it is even possible to interpret %/Yas an algebraic ring4of operators
in a suitably defined complex Euclidean space I whose dimension number is the
cardinal number of the c o n t i n u ~ m . ~ to put it differently, there exists in
the ring of bounded everywhere-defined operators over I a subset % which is a
(+, . , *)-isomorphism of 58/T Furthermore, this correspondence is even an
isometry, the norm of an element of 58/3being the bound of the corresponding
element of W. These facts are all established in 554, 5, and in 55, various prop-
erties of the algebraic ring % are discussed.
   Apart from its intrinsic interest, the analysis of the ring % yields in a very
simple way theorems of considerable depth concerning 58 (55). Of results of
this sort, we shall here mention only a generalization of MTeylJs  classical theorem
comparing the spectra of two self-adjoint operators with totally continuous
dfference [ 2 0 ] . ~
   Before proceeding, we should like to point out various further developments
which are suggested by the present paper and to which we propose to return a t
another time.
   First, while the algebraic ring W is not closed with respect to the weak
topology for operators and so is not an operator ring in the sense of von h'eumann
[9, 111 one obtains such a ring by adjoining to % its weak condensation points
in the ring of bounded everywhere-defined operators in I. This "closure"
R(%) is of considerable interest from the point of view of the Murray-von Xeu-
mann theory of factors, especially since various preliminary results concerning
R(%) and a' suggest that they may be factors of class III,.'
   Second, while the factors of class 111and 111, of Murray and von Seumann
are like those of class I,, n < No,  simple rings, those of class 11, are not. The
construction of quotient rings over such a factor is therefore possible and gives
rise to various questions analogous to those concerning 9 with which we deal
   Finally, we should like to point out that the maximal property of the ideal 3
of totally continuous operators described above does not persist if one considers

   'We use the term algebraic ring with reference t o operators t o denote a class closed with
respect t o the operations   +, . , *, a n d scalar multiplication. T h i s is not a ring in t h e sense
of von Neumann, [ l l ] , since n o topological conditions are imposed.
     For the theory of complex Euclidean spaces of arbitrary dimension number, see refer-
ences I61, 171, [131, MI.
   "1 numbers in brackets refer t o the bibliography a t t h e end of the paper.
     For the notion of factors, and their classification, see [9]; for a construction of factors
of class 111,, see [15].
     T h e class of members of a 11, factor which are "normed" in t h e sense of [15] is a two-
sided ideal.
                      BOUNDED OPERATORS IN HILBERT SPACE                               841

instead of Hilbert space, a space Q with dimension number greater than $40.
For example, in such a space, the class of all operators A such that X ( A ) contains
no closed subspace with dimension number exceeding No is a non-trivial ideal.
Moreover, if m is the dimension number of @, m >= No, the class of all o p e r a t ~ r s
A which have the property that % ( A ) contains no closed linear subspace of @
with dimension number m is a two-sided ideal different from 9and is identical
with Tif m = No. Furthermore, in every case, this ideal is maximal in the same
sense in which Tis for m = No, andas von Seumann has pointed out to the author,
the spectral characterization of ideals can be extended in a satisfactory way
to describe the general case. Finally, investigations which are still incomplete
suggest that a considerable portion of the analysis of the present paper has a
counterpart in every case.

                                  1. Two-sided ideals in 5   3
   We proceed now to the characterization of all two-sided ideals in 9.
   THEOREM      1.1. I f 9zs a left (rzght) zdeal, the set 9* of all adjoznts A * of elements
A of 9zs a rzgl~t( l ~ j t ideal.
   Since g* is evidently closed with respect to addition, it is necessary only to
show-when 9is a left ideal-that g* contains B A * for all B in 3 and -4 in 9                .
But this follows a t once from the lelation B A * = (dB*)*and the fact that A B *
belongs to g along with A. When 9 is a right ideal, an analogous argument is
   THEOREM A necessary and sufiezent condition that a left (rzght) zdeal 9
be two-sided i s that 9 = g*.
   The sufficiency of the condition is an immediate conscqucnce of Theorem 1. l .
Now suppose 4 i s two-sided, A an arbitrary member of 9, -4 = JYB its canonical
d e ~ o m ~ o s i t i o nThen A * = B W * = TV*AW* is clearly in 9 and 9 = 9*.
   Since we now have no further direct concern with left or right ideals we shall
refer to two-sided ideals merely as ideals.
   THEOREM T h e class T of all totally contznuous operators zn 9zs a n zdcal.
   That Tis an additive class is obvious from the definition of a totally continuous
operator; 1' is totally continuous if it takes every bounded set into a compact
set. Moreover, by a well-known theorem, T = T*.1° Finally, since every mem-
ber A of $8 clearly takes bounded sets into bounded sets, 3 i s a left ideal. Hence
by Theorem 2 it is an ideal.
   Throughout the remainder of the paper T has the same meaning as in The-
orem 1.3.
   THEOREM Let g be a n arbitrary zdeal in $8. T h e n ezther 9 = 9or 9 6;; 3.
   The proof of this theorem is based on a characteristic property of totally
continuous operators which the writer has noted elsewhere                     -4ccording to

    For the notion of canonical decomposition, see [Q] and [12]. 

     [2], p . 100, ThBor6me 4 . 

  I1 (31, Lemma 3.1. 

842                                      .
                                        J W. CALKIN

thls result a member T of 3 is total!y continuous if and only if every closed
linear manifold in its range has a finite dimension number. Hence, if $is a n ideal
which is not contained in 3; 9 contains an element A such that % ( A )contains a
Hiibert space 8 . We denote by 92 the manifold of zeros of A and by A1 the
transformation induced on Sj Q 9, by A,'' The transformation Al evident'ly
possesses a n inverse and the same range as A ; moreover the fact that Al is
bounded assures us that the set ? Ac'S% is closed. Thus $1is also a Hilbert
space. Hence there exist in 9 partially isometric                        X , Y both with
initial sets @, while % ( X ) = 3, % ( Y ) = S%. Therefore the operator B =
Y*AX has domain and range identically @ and belongs to 9.But B f = 0
implies either X f = 0, X f in 97, or A X f in @
                                               .     !JX, and all of these are impossible
in view of the definition of X and Y . Hence B-' exists and since it is closed
with domain @, it belongs to 9.Therefore I = B-'B hclongs to $ a n d g = 9.
   THEOREM     1.5. Let 9 be a n arbitrary ideal in $ 3 9d the class of nonnegative
dejinite self-adjoint transformctions i n 9. T h e n goi s the class of all operators B =
(A*A)' such that d i s a n clement of 9, and if 9' i s a n ideal containing $0, then
91 2 9.
   If B is in g o ,it is obvious that B is of the form described in the theorem.
On the othcr hand, if A is an arbitrary element of 9, (A*A]' = B = W * A ,
\r.hcre R is partially isometric; hence ( A * d ) $ belongs to 9, and thus to -90.
Finally the relations d = W B , B = ( A*A)' assure us that 9is the smallest ideal
containing g o .
   If 9is an ideal in 53,we call the subset godefined in Theorem 1.5 the positive
liart of 9. In order t:, characterize those subsets of the positive part of 3
which appear as the positive parts of ideals, we recall that every self-adjoint
operator ?' in T c a n be rcduced to diagonal form; that is, for each such T there
exists a complete orthonormal set { p n j in @ and a scquence { A , / such that
Tpn = Anp, , n = 1 , 2, . . . . Moreover, the sequence ( A , ) is convergent to
zero, and nonnegative if 7' is in the positive part of tT Jf7e call this sequence a
characteristic sequence of T .
   We next observe that if 1 belongs to the positive part of an ideal 9and I$,}
is another complete orthonormal set in @, thrn the member A of 3dcfined by
the equations A+, = A,$, , n = 1 , 2, . . . , also belongs to the positive part of 9,
sicce A = C T I ' - ' , where C is unitary. Hence, if 4 is a n arbitrary ideal in 3,
g 2 3the set of all characteristic sequences j A, / belonging to members of 90
may without ambiguity be called the spectral set of 9.
   We now characterize intrinsically those subsets of the class of nonnegative
sequences with limit zero which occur as spectral sets. As is indicated in the
introduction, this result is due t'o J. v. Seumann.
   DEFINITION . 1. Let 5 denote the class of all injinite sequences of nonnegative

  l2   We regard A as a transformation between Hilbert spaces in the sense of [ a ] . 

  l3   For this c?ncept, see [ 9 ] ,Definition 4.3.1. 

                       BOUNDED OPERATORS IN HII~BERT SPACE                                     843

 numbers which co7zverge to zero. -4 subset 3 of 2 i s called a n ideal set i f i t has the
following properties:
     (i) I f (A,) i s in 3 a n d ?r denotes a n arbitrary permutation of the positive integers,
 j A r ( n , ) i s in 3;
     (ii) if {A,) and {p,) are in 3, so i s {A,   +    p,) ;
     (iii) if {A,) i s in 3, (p,) in % and the inequality A, 2 p, holds for all n, then
 {p,) i s in 3
     Our object now is to prove that every spectral set is an ideal set, and con-
 versely. We require first two lemmas concerning ideal sets.
     LEMMA          1.1. Let 3. be a n ideal set, (A,) a n element of 3 w i t h injinitely m a n y
 terms different from zero. T h e n the subsequence of positive terms of {A,) belongs
 to 3.
     We distinguish two cases according as (A,) contains a finite or an infinite
 number of zeros. In view of condition (i) of Definition 1 , we can assume in
 the first case A1 = A2 = . . . = AN = 0 and 0 < An+1 5 A n , n = N                  +   1, N   +
 2, . . . . We then have to show that                     belongs to 3. Let (p,) be the se-
 quence defined by the equations

Then ( f i n ) is dominated hy a permutation of (A,) and hence belongs t o 3, by
conditions (i) and (iii) of Definition 1. Moreover, by condition (ii), (A,          pn)    +
belongs to 3. But we have AN+, 5 An              +
                                            pn , n = 1 , 2, . . . , and hence { A N + , ]
belongs to 3.
  Now suppose (A,) contains an infinite number of zeros. Again invoking (i),
we assume

We must then show that         (Atn)   belongs to 3. We define a sequence           {p,)   by the

Then (p,] is a permutation of (A,) and hence belongs to                 3.    But then (A,      +
p,]belongs to 3, and we have

Thus, by (iii), ( A t , ) belongs to 3 as we wished to prove.
  LEMMA     1.2 Let 3 be a n ideal set, (A,) a n element of 3 consisting solely of
positive terms. T h e n a n y sequence (p,) which contains {A,) a s a subsequence a n d
which, except for this subsequence, consists solely ojzeros, i s a member o j 3 .
  R e define
844                                   J. W. CALKIN

Then {A?') and          are both dominated by (A,) and therefore belong to 3.
Letting j k denote the integer immediately preceding k/3, we now define two
permutations a l l an by the equations,

The effect of the first of these pernlutations ib to take the set of all integers divisi-
ble by 4 into the set of all even integers, prc-erving order; and to take the set of
all integers not divisible by 4 into the set of all odd intrgers, again preserving
order. The effect of the second is to take all even integer5 not divisible by 4
into the set of all odd integers, and the complementary set of integer< ints the
set of all even integers, order being preserved in hot11 cases. Hence the sequence
                               (vn)   =   (A(,:)(,)   + A?;;,)i
which clearly belongs to 3, is related to the sequence (A,) by the equations

But any sequence ( p , ) derived from ( A , ) in the manner described in the theorem,
and containing an infinite number of zero terms, is a permutation of ( v , ) and
therefore belongs to 3.
  Thus, to complete the proof, we have only to dispose of the case t h a t (p,}
contains only a finite number of zero terms. In this case it is convenient to
assume that {A,) is monotone. We can then complete the proof by showing
that the sequence { p , ) defined by the equations

belongs to   3. T o do this we set

Then, from the validity of the lemma in the case of an infinite number of zeros
and property (ii) of ideal sets we can conclude that { v , ) is in 3. But, since we
clearly have

it folloivs that { p,) is in 3.
   THEOREM Let 9 be a n ideal in 3,9
               1.6.                                   s
                                                     and let 3 be its spectral set. T h e n
3 i s a n ideal set. Conversely, i f 3 i s a n arbitrary ideal set in %, there exists a n
ideal in 3 whose spectral set i s 3. T h i s correspondence between ideals in $3 and
ideal sets in X i s a n isomorphism with respect to the relation C .
   If 9 is a two-sided ideal in $3,9  s      we consider a n arbitrary member A of 90
in diagonal form; Ap, = Xnpn, n = 1, 2, . . . , where (p,) is a complete ortho-
normal set in @. Then (A,} is in the spectral set 3 of 9, since A can also
                      BOUNDED OPERATORS IN HILBERT SPACE                                        845

be put in diagonal form with reference to any permutation of the sequence
{p,), all permutations of (A,} clearly belong to 3 too.
  Xow let B be any other member of 9 0 , also in diagonal form; B+, = p,+, ,
n = 1 , 2 , . . . , and let II be the unitary operator defined on (p,} by the equations
Up, = +,, n = 1 , 2 , . . . . ThenA       +   ~-'~Uisin9~and
                (A   +u-'Bu)~,        = (A, + p,)p,,              n    =   1, 2,   . . ..
Thus (A,   +  1,) belongs to 3.
  Finally, let (p,) be a sequence such that     p,   5 A,, n           = 1, 2,     . . . . We define
an operator Bo on (p,) by the equations
                            Bwn   =   (~nlAn)~n
                                             1           An   f   01
                            Bw,   =   0      ,           A,   =   0.
Then Bo has a closed linear extension B which belongs to 9. Thus AB belongs
to 9. But ABp, = p,p,, n = 1, 2, . . . , and therefore AB belongs to go,
 ( ~ n to 3.
    We turn now to the converse part of the theorem, denoting by 3 an arbitrary
ideal set in 2. We designate by 9 the set of all totally continuous operators A
such that a characteristic sequence of (A*A)' belongs to 3. Since, if 9 is an
ideal, 3 is obviously its spectral set, we have only to show that 9 is an ideal.
    We begin by considering an arbitrary pair of operators A and B of 9, with
the object of showing that A      +B belongs to 9. To this end we consider the
         Dl   =   (A*A)',    Dz   =   (B*B)',        D   =    (A*      + B*)(A + B)'
and characteristic sequences (A,}, ( p , ) , (v,) of Dl , Dz , Dl respectively. We
have then to show that (v,) belongs to 3.
   For convenience, we assume that the positive terms of each sequence are
arranged in montone order; in addition, if any one of the sequences contains
only a finite number of positive terms, we assume that the sequence itself is
monotone. This assumption is clearly not restrictive, in view of condition (i)
of Definition 1. Now let (A:) be identical with the subsequence of positive
terms of (A,} if that subsequence is infinite, identical with (A,) itself in the altern-
ative case, and let (p:) and (v:) be defined in the same way with reference to
{pn) and (v,} respectively. Then, by Lemma 1.1, (A:} and (p:} belong to 3;
and by Lemma 1.2, (v,} belongs to 3 if (v:) does. Thus we can prove that
A  +  B belongs to g by showing that (v:) belongs to 3.
  To establish the latter result, we note first the relation
        2(A*A  + B*B) - (A* + B*)(A + B) = (A* - B*) ( A - B),
which implies that 2 ( ~ :+ 0:) - D~ is nonnegative definite. Holding this
fact in reserve, we then recall the theorem of Courant [4],14characterizing the
  l4 The paper cited deals only with integral operators with continuous kernels.              How-
ever, the more general result required here is readily obtained.
846 	                                                     J. W. CALKIN

sequence { v ~ associated with the operator D2 in terms of maxima and minima.
                ~ )
For our purposes, we may state this result as follows:
                                   vn2 =                 Min                   Max (D2j,j)
(1) 	                                             dim (lDl)sn-1         f c SjeW

                                                                                    If    -

where dim (8l)is the dimension number of 9,R. Similarly,
                               Xn =                      Min                               ,
                                                                               Max ( ~ l jj),
(2) 	                                             dim (W)Sn-1              f    t        48W 

                                                                                If        -
                                                                                         1 1

                               pn =                      Min               Max (Dij, j).
                                              dim        (9)Sn-1       f       t       488
                                                                               If         -

On the basis of these relations we are able to conclude that the inequality

is valid provided we have j     k             + s n + 1.                                        For if 	 v
                                                                                                         :      >   2(xj2   +~      )
                                                                                                                                 2 holds,
we have, in view of (2) and (3),
         vL2/2 >           Min
                     dim (%)Si-1 f e
                                             Max (Dlf, j)
                                                                                +         dim
                                                                                                   Min          Max (D: j, j)
                                                                                                   (B)6k-l f c 489
and this implies
            vL2/2 >
                              (W+gl)Si+k-Z [,Max (Dl(, f )
                                              e QgW
                                                                                                    +    f
                                                                                                             Max (D: j, j)]
                                                                                                             * QeB
which in turn implies
                 2        >  Min
                           (%+B)Si+k-l[f e

j being restricted in every case to satisfy I j I = 1. But then, since D:                                                        + D:   -
D2/2 is nonnegative definite, we have
                               :         >               Min                    Max (D% j)
                                             dim (!TI?) S j f k - 2        f       t      D
                                                                                         Ql l
                                                                                    If    -

which contradicts (1) unless j                    +
                               k - 2 is greater than n - 1, or unless j    k                                                        +
exceeds n   +
            1. Thus (4) holds for j   k 5 n                      +
                                                 1, as stated, and we have                         +
(5) 	                                                VL    5               + PL)     forj+kSn+l.
  Now let rn be n/2 if n is even, (n                        +    1)/2 if n is odd. Then, from (5) we have
(6)                                                v:,    5 2(C,           + P:,)

Moreover, the sequences (A:),                            (/,'I defined by the equations 

                                                         = 0,              X2k
                                                                                           = XL                         k = l,2,    ... ,
                                                                                           = CL:                                    ...,
                                             It                                     11
                                         P2k-1              0,             P2k                ,                         k = l,2,
belong to   3 by Lemma 1.2; and since
            A:,, =
                           +    tt
                               Xr(n)                     k. =      Pn
                                                                               +              1
                                                                                                    ,           = 1, 2,     ..
                        BOUNDED OPERATORS I N HILBERT GPACE                             847

where r ( 2 k - 1) = 2k1 7 1 2 k ) = 2k - 1, it follows from conditions (i) and (iii)
of Definition 1.1 that the sequences {A:,] and (p:,,) both belong to 3. But
then, by (6) and conditions (ii) and (iii) of our definition, (v:] belongs to 3)
and this completes the proof of our assertion that A        B belongs to 9.     +
   Xow let A be an arbitrary element of 9, Dl = (A*A)" X an arbitrary element
of 9, D: = (A*x*xA)'.            Since A is in 3; X A and 1 9 2 are also. Let {A,],
      { A: 1, 1 fik ) have the same meanings as above with reference to Dl and D .
Then (A: ) is in 3 and we can show that X A is in 9by showing that (p: ] is in 3 ;
the arguments here are the same as above.
  T o establish the latter, we again apply the theorem of Courant;
                             A 1,2=           Min                 Max ( ~ : f , f ) ,
                                        dim (Dl) s n - 1      /   e
                               f   2
                             p n =            Min                 Max @if,!).
                                        drm t 9 n ) S n - 1   / r WR9
                                                                 1 11

But ( ~ f f ), = (Af, Aj), (D~I, j) = ( X A f , XAB and ( X A f , X A j ) is bounded
by N 2 ( ~ jAf) for some integer N. Thus we have

and a s ( N X ~ )clearly ir. 3, so also is { p k ) . Hence XA belongs to 9
                is                                                               .
   Thus 9is a !eft ideal and in view of Theorem 1.2, we can show t h a t g is two-
sided by showing that ,4 is closed with respect t o the operation *. But this is
an immediate consequence of 1,ernmsis 1.1 and 1.2 and the well-known fact
that (A*A)' and ( d r l * ) ' have the same positive characteristic values, each
with the same n ~ u l t i ~ l i c i t y . ' ~
   The concluding assertion of the tht.orem is obi'ious.   '

   THEOREM     1.7. Lrt 7 denote the class of all operators A zn $8 such that %(A)
has a finzte dzmenszon number. Then 3 LS a two-sided deal en 9. If 9i e an
arbitrary two-szdcd zdcal, 211 %, thcn 9 = (0) 9 2  or
  T o establish the assertion, we have only t o note t h a t the class 8 of all
sequences in 5 with only a finite number of terms different from eero constitutes
an ideal set. T o establish the second we first observe that any ideal set different
fmm the one containing only the iequencc all of whose terms are zero, contains
a sequfnct I A, ] n i t t i A, # 0, A, = 0 fur 72 # 1 , by virtue of (i) and(iii). Hence,
by (ii) and (iii) it contains all sequences of this sort and hence, by (i) and (ii),
contains    s.
   I t is worthnhile to observe here that the effect of Theorem 1.6 is to establish
an isomorphism between the lattice L1 of two-sided ideals in 9 and a ccrtain
sub-lattice L2 of the lattice of idtlals in the ring 8 of all bounded sequences of
complex nurnbcrh. To makc this clear n e require two preliminary results
concerning %; the first theorem is the analogue for 23 of Theorem 1.5 for 9 the
second is analogoui to Thcorcms 1.3 and 1.1.

  Is   By [12], Satz 7 , for example.
848                                      J. W. CALKIN

     THEOREM       1.8. Let 8 be the ring of all bounded sequences of complex numbers,
3 a n ideal in 23. Let 3o   be        the set of all sequences ( 1 X , / ) such that ( 1,) i s in 3.
T h e n 3: contains 30 and i f           i s a n ideal in $3 containing      , then g1 2 3.
     The proof is straightforward and is left to the reader.
     THEOREM       1.9. Let L2 be the lattice of all ideals 3 in 8 which satisfy the fol-
 lowing condition; if ( a , ) i s in 3,        and t i s a permutation of the positive integers
  (a,(,,] i s i n 3 . T h e n tlze set 'Z of sequences convergent to zero belongs to Lz , a n d i f
3: i s a n arbitrary element of L 2 , either 3: = 8 or 3 2.
     The first assertion is obvious. Now if 3 is a member of Lz which is not con-
 taincd in 5,3:contains a sequence ( a , ) which has a subsequence ( a,,] such t h a t
  ( a ; : ) is bounded, and which converges to a number different from zero. Fur-
 thermore, the sequence {b,) with b,, = a,, , b, = 0, n # n k can be written

  (b,] = ( c , ) ( a , ) , where c,, = 1, c, = 0, n # nk , and hence {b,] belongs to 3 .
 \Ye now write the sequence ( b,] in the form { b, )               pne'8"), p, > 0, and observe
 that {p,l also belongs t o 3 by Theorem 1.8. But then by the same sort of
 argument that was used to prove Lemma 1.1, we can show t h a t ( p , , ) belongs
 to 3 , and hence ( a n , ) does also. Hence (el;) = (a,,] (a,:) belongs to 3
 andek = 1, k = 1 , 2 , . . . . T h u s 3 = 23.
     THEOREM 1.10. Let 3 be a member of L2 , gothe subset of 3 defined in Theorem
 1.8. T h e n either 3 0 = 8 0 , where B0 the set of all sequences of nonnegative numbers
 i n 8,or every sequence in 30 s convergent to zero and 30 s a n ideal set in the sense
                                      i                               i
 of De$nition 1.1.
     THEOREM       1.10 follows a t once from Theorems 1.8 and 1.9.
     THEOREM       1.11. Let L1 be the lattice of all two-sided ideals in 9 , let the set 230
 be called the spectral set. of '3, and let the class of ideal sets be extended to include
 2 0 . T h e n L1 i s lattice-isomorphic to the extended class of ideal sets, each member
of L1 corresponding under this isomorphism to its spectral set. S i m i l a r l y , the
lattice L2 i s lattice-isomorphic to the extended class of all ideal sets, each member
3 of Lz corresponding to its 30. T h u s L 1 and Lz are lattice-isomorphic a n d under
this isomorphism 9 corresponds to 8 , 9- to the class 2 of all sequences convergent
to zero, y t o the class 8 of all sequences with only a finite number of terms different
from zero.
     Theorem 1.11 is obvious on the basis of preceding results and we omit the
proof. Before proceeding, however, we wish to make the following observations:
The ring $3 can be imbedded in 9 in a very simple way; we have merely to
choose a complete orthonormal set ((p,) in @ and identify the element ( a , ) of 23
with the closed linear operator A in @ which is defined on ((p,) by the equations
Apn = an(pn , n = 1, 2, . . . . Moreover, if we consider 8 in terms of th'is identi-
fication, each two-sided ideal 9 in $8 corresponds under the isomorphism of
Theorem 1.11 merely to its intersection with 8 . This suggests a n alternative
attack on the problem solved by Theorem 1.6; however, in so far as we can
determine, it is not possible to devise any essentially different proof of t h a t
theorem on this basis.
                           BOUNDED OPERATORS IN HILBERT SPACE                              849

                                 2. Congruences in 3
    We pass now to the study of congruences modulo an ideal 9 in 9. Following
the standard procedure of abstract algebra, we consider the class 3/9 whose
elements a, p, . . . are the residue classes of 9 with respect to 9 ; by definition,
two merrlbers A and B of 9 belong to the same element a of %/$ if and only
if A - B is in 9. If a and P are arbitrary elements of 9/9, we define a                   +
as the class of all elements A        +
                                    B of 9 such that A is in a, B in P ; similarly, we
define a@ the class of all A B in 9 such that A is in a , B in P. Then,fromthe
general theoremI6 which is controlling in such situations, we have
    THEOREM                                                              +
              2.1. I f a a n d P are elements of 9/9, so also are a P and aP. W i t h
addition and multiplication defined in this w a y 9 / 9 i s a ring; that i s to say, 9/9
i s a commutative group with respect to t h operation
                                               ~               +,
                                                             and further, the following
formal laws are satisfied:

iMoreover, 3/9 possesses a u n i t .
   I t may be noted that except in the case 9 = 9, the subclass of 9/9 each of
whose elements contains a scalar multiple of the identity in $8, is isomorphic to
the class of scalar multiples of the identify in 9, since no two of these elements
of 9 can have difference in gunless they are identical. I t is convenient therefore
to use italic letters for these elements as well as for the corresponding elements
of 9.In addition we shall use the symbol 1 for the unit in 9/9; that is, for the
element of 919 whose members have the form I                 +
                                                           T , where I is the identity
in 9and T belongs to 9.
   I t is worth pointing out here that the ring 9 / 9 , 9 # 9, is certainly non-
commutative; to verify this one needs only to consider two orthogonal projec-
tions E and F whose ranges are Hilbert spaces and whose sum is the identity,
a partially isometric operator W which maps E on F, and the operator W*W -
W W * = E - F which belongs to no ideal except 9 itself. Later we shall show
that the center of 9/9, 9 Z 9,is the set of all scalar multiples of unity (Theorem
   THEOREM     2.2. I f a i s a n arbitrary element of 9/9, the class a * of all members
A* of 3 such that A i s in a i s in %/9 also. T h e operdtion * so defined in 9/9
obeys the following laws:
                  a**   = a,       (a+P)*=a*+P*,                 (a@* = @*a*.

  That a* is in 9/9follows a t once from the fact that 9* = 9 ; and the three laws
stated in the theorem are readily verified on the basis of their validity in 9 .
  Thus we see that the rings 9 / 4 have all of the formal properties of matrix
algebras and are homomorphs of 9with respect to the operations                   +,
                                                                          . , *, a . ;
  16   [I], pp. 252-253.   The missing details necessary for our purposes are readily supplied.
Cf. the discussion of the commutative case in [19].
850                                J. W. CALKIN 

moreover, they are of course the only homomorphs of % with respect to these
    I t is now desirable to consider these homomorphisms with respect to the fol-
lowing important notions in operator theory; for an operator to be self-adjoint,
to be idempotent, to be partially isometric, to be unitary. Hence we are led
to define these concepts in 9/9 without explicit reference to their meanings in 9.
    DEFINITIONAn element a: of 9/9 i s called self-adjoint if a = a*; a self-
adjoint element e of % / 9 i s called idempotent if = e; a n element o of %/4i s called
partially isometric i f w*w = e i s idempotent, unitary i f w*w = ww* = 1.
    It is easy to see that under the homomorphisms % -+ % / 9 the image of
every-self adjoint operator is self-adjoint and that analogous assertions hold for
projections, partially isometric operators, and unitary operators. We shall
now show that with reference to the first two of these concepts the converse
statements are also true.
   THEOREM I f a: i s a self-adjoint element of 9/9, a: contains a self-adjoint
member of 9, and conversely.
   Let a be self-adjoint, A an element of a . Then A* - A is in 9 and hence
A  +     (A* - A)/2 = (A   +       A*)/2 is in a. The converse, as we have already
noted, is obvious.
   THEOREM.4. I f e i s a n idempotent element of %/9, there exists a projection
E in % which belongs to e, and conversely.
   The theorem is obvious for 4 = $8; we assume therefore 9 3: By Theorem
2.3, e contains a self-adjoint transformation A, and since e is idempotent,
A' - A = A(A - I ) is in 9 and thus in 5: Hence A' - A can be reduced to
diagonal form, and therefore A can also; Apn = h n p n , n = 1, 2, . . . , where
f V n ) is a complete orthonormal set in @. Rut then it follows that {A,] contains
a subsequence (A:')      convergent to zero, and such that the remaining terms of
{A,) form a subsequence, say {A:'],          convergent to 1, since under any other
circumstances A(A - I) would fail to be totally continuous. Moreover, we
can clearly assume that {A:']        contains no terms with the value zero and that
(A?') contains no terms with the value unity.
   Now let 9Jlo be the subspace of @ determined by the characteristic elements
of A corresponding to terms of {A;'], !Dl1the subspace determined by the other
characteristic elements of A , E and El the projections with ranges %Qo and % Q l ,
respectively. We shall show that El belongs to e. To do this we note first
that in m0,     A - I induces a transformation with bounded inverse. Hence if
B is equal to this inverse in !Dl0 and to zero in m 1 ,
                            E ~ ( A ' - A)BEo   =   EoAEo
is in 9, since A' - A is. Similarly, it follows that
                          El(A - I)E1    =   ElAEl - El
is in 9. But then, adding the right members of the two preceding equations
we find that

                        EoAEo   + ElAEl     -   El = A - El
is in 9, and from this it follows that El belongs to E .
   Again the converse part of the theorem is obvious, so the proof is complete.
   THEOREM    2.5. Let 9 be a n zdeal in $8, go the sct of all nonnegatzce deJntte
self-adjoznt elements of 9, 9;the class of all squares of elements of 90. T h e n a
necessary and suficzent condztion that ccery partzally zsom~trzcclt ment w of 9,   g
contazn a partially zsometrzc transformatton W t s that $0 = 9; .
   Again the case 9 = $8 is trivial, so we assume 9 2 T. Let w be partially iso-
metric, V an element of w, 1 = 17R its canonical deconlpohition. Then
V*V = BZ belongs to an idempotent element 6 of % / 9 . Hencc, if            identify
BZwith the self-adjoint transformation A which appears in the proof of Theorem
2.4, we can invoke that thcorem to e5tablish the existence of a projection E
such that B2 - E is in 9. JIoreover, an inspection of the proof revclals also
that BZ commutes with E and induces rn the range of E a transformaticpn with
bounded inverse. I n particular, this implies that E has for its range a subspsce
of the initial set of [ T and thus that ['E is partially isometric, since
E17*C7E= E." We shall now show that under the condition of the theorem
B - E is in 9. JVe note first that since BZand E commute, \nc have B2 - E =
( B - E ) ( B E ) , and since B i i nonnegative. B    +E induces in % ( E ) a trans-
formation with bounded inverse. Hence, if C is equal to this inverse in % ( E ) ,
and to zero in @ 7 % ( E ) ,we have ( B Z- E)C = EB - E l and EB - E is in 9
                    .                                                               .
hloreover ( I - E)(B' - E ) = ( I - E ) B ~ in 9. But if 4 has the property
described in the theorem, [(I- E)B']' = ( I - E ) B is in 9, and thus
is in 9as we wished to show. Rut then V - 17E = I'(B - E ) is in 9and
W = I7E1     which we have alrcady shown to be partially isometric, bclongi to w .
   I t remains therefore for the converse part of the theorem to bc provcd T o
this end, we suppose that 9 contains a nonnegative definite .elf-adjoint trans-
formation BZsuch that B does not belong to 9, and denote by w the congruence
class in 3 / 9 to which B belong< Then, since B*B = B? belong.; to ,I       ',   ha\ e w
partially isometric by definition. But if V is a partially i\omc.tlic tran.forina-
tion in w , V*V = E must be congrucmt to B? modulo 9, n hlcli i- to .a>. tllat E
is congruent to zero modulo 9. Hut thii implies that E has r:Liigo \\it11 hnlte
dimension number and the range of E is the initial set of V . T h u i V 15 in
Yand hence in 9. However, V - B is in 9since T7 is in w , and as B 151,- aiiump-
tion not in 9, have a contradiction. He~lc-c, condition of the. thcorcm is
                 we                                 the
necessary as well as sufficient.
   R e may note in passing that the ideals (O), ff, ,tl; CR all satisfy the condition of
Theorem 2.5, but that these are not the only idral.; ~ \ h i c h so. ('on>idrr, for
example, the class of all sequences jX,1 of no~lnegative numbers sucali that

    By [9], Lemma 4.3.2.
      X,P converges for some p.   I t is easily seen t h a t this class is an ideal set in the
sense of Definition 1.1. and t h a t the corresponding ideal in B has the property
of Theorem 2.5.
   For the sake of completeness, we state the following obvious theorem:
   THEOREM I f W zs a partzally zsometric member of 3, the congruence class
of W zn 31.9zs partzally zsometrzc zn $ 1 .
   THEOREM 7. Let 9be a n zdeal zn 93. T h e n , z j w i s a unztary dement of %/9,
w contazns a mnxzmal partzally zsometrzc t r a n ~ f o r m a t i o n 'wzth deficzency-zndez
(0, n ) or ( n ,O), n < No.
   Agaln the cabe 9 = 93 is trivial, so we assume 9 # 9. Let w be unitary, V a
member of w, V = IqB its canonical decomposition. Since w is also partially
isometric, the first part of the proof of Theorem 2 5 applies to yield the follouing
results therr exists a projection E with range in the initial set of C , and uhich
commutes u i t h B, such that V*V - E = B? - E and E B - E are in 9. But
since V*V is congruent to I modulo 9, I - E is in 9 and thus ( I - E ) B is in 9.

is in 9. But then V - I'E = l ' B - I'E is in 9. Hence W = Z'E is a partialiy
isometric operator which belongs to w . hlortover, since u is unitary, I - WW*
is in 9 and hence, since this operator is a projection and belongs to 9, its range
must have a finite d i m e n d m number. Similarly, I - W*W has range with a
finite dimension number. Thus both the initial and final sets of W h:ive or-
thogonal complements with finite dimension numbers. T!~erefore, if TITl is the
contraction of W with domain % ( E ) and X a maximal partially isometric exten-
sion of Wl , X has the property required in the theorem and X - W is in 9.
Thus X belorigs to w, and the theorem is proved.
     I t is important to observe t h a t every unitary element of 93/9 does not contain
a unitary member of $8, except in the trivial cases 9 = 93, 9 = (0).
     T o prove this, we consider an isometric transformation X with deficiency-
index (0, n ) , n < &, and the congruence class u modulo 9, to which X belongs.
Then LO is clearly unitary in !3/9provided 9 # (0). S o w suppost C 7 is a unitary
transformation in LO. Then l' - X is in 3; and thus in tT, if 9 # 9 H t n c t I -
['-'A7 is in 9and I'-'x also has deficiency-index (0, n ) . But by a lemma which
the author has proved elsewhere, this is possible if and only if n = 0 l9
     THEOREM                                 3
                2.8. Let 9 be a n zdeal in 9 dzflerent from (0), U a partzally isometrtc
operator i n $8 with deficiency-index (m,n ) m, n < No.     7'1z~n thecongruence class w
i n % / 9 to which W belongs i s unitary.
     If W has the properties stated then I - W*W and I - W W * are projections

   l 8 We call a partially isometric operator maximal if the isometric transformation which
determines i t is maximal. Similarly, we shall have occasion t o refer to the deficiency-
index of a partially isometric operator.
   l 9 [3], Lemma 4.1.
                     BOUNDED OPERATORS IN HILBERT SPACE                            853

which belong to X Thus, since 3 g 4 by Theorem 1.7, both of these operators
belong to 4 and o is unitary by definition.
   We conclude this section with
   THEOREM Let 4 be an ideal in 9,9 # 9. Then the center oj 914, that is, .
the set oj all elements oj $314 which commute with every element of 914, is the set
oj all elements A. 1, wkre X is a complex number.
   I t is clear that the center contains the set of all scalar multiples of the identity;
hence we need only show that it contains no other members.
   We begin by showing that it is sufficient to consider merely the self-adjoint
members of the center. For suppose CY belongs to the center. Then a@* -
P*a = 0 for all /3 in 9 1 9 and hence (a@*- @*a)*= Pa* - a*@= 0 for all P
in 914. Thus a* belongs to the center, and consequently the self-adjoint
elements a  +      a* and i ( a - a*) do also. Now suppose a     +    a* = A . 1, i ( a -
a*) = p . 1. Then, eliminating a*, we have a = ( p iX)/2i. Hence we have
only to prove that every self-adjoint member of the center is a scalar multiple of
the identity.
   In terms of operators in 9, this problem reduces to the following: to show
that every self-adjoint operator A in 9 such that AB - BA is in 9 for all B
in 9 is of the form T XI, where T is in 4.
   We consider first the case 9 = 3 So we consider a self-adjoint operator A so
that AB - BA is totally continuous for all B in 9. If A is not of the form T        +
XI, T in T, the spectrum of A must contain two distinct points, each of which is
either a limit point of the spectrum of A or a characteristic value of infinite multi-
plicity; for, otherwise, the spectrum of A consists solely of isolated characteristic
values of finite multiplicity together with one point p which is either a limit point
or a characteristic value of infinite multiplicity, and in this case A -          clearly
belongs to 3. Hence if E(X) is the resolution of the identity of A , there exist
numbers $O X1 , X2 , X3 in the spectrum of A, Xo < X1 < X2 < X8 such that E(X1) -
E(X0) and E(X3) - E(X2)have ranges Y J l and Y J 2 , respectively, which are Hilbert
  Next let us consider the partially isometric operator W with initial set YJl
and final set YJ2 . We then have, for j in !?XI
                      I W A j - AWjI 2 IAWjl - IWAjI,
and thus, since we also have

we obtain
                          W A j - AWj I 2 (At - k ) j .
Hence ( W A - A W ) induces on %Vl a transformation with bounded inverse and
therefore ( W A - AW)YJl is a Hilbert space. Consequently, by a lemma
previously referred to, W A - AW is not in 5. Hence the assumption that A
is not of the form T AI, T in is untenable, and the theorem is established for
854                                    J. W. CALKIN

     Consider now an arbitrary ideal 9, F C 9 C                 9 # 3, g # Z As before,
we consider a self-adjoint operator A such that A B - B A is in 9 f o r all B in 9.
Since 9 C 3; it follows from the preceding result that A is of the form T                +
X I , T in EK Thus TB - BT is in Sfor all B in 9. Now let {cp,) be a complete
orthonormal set of characteristic elements of T ; Tpn = Xnpn, n = 1, 2, . . . ,
and let us suppose that T is not in 9. Moreover, let us assume that the sequence
{p,] is so arranged that X1 is different from zero, while the positive terms of
 ( / A, / ) are in monotone order. Let (p,) be a sequence of positive numbers in
the spectral set of 9 with pl < / X1 1; further let (A,,) be an infinite subsequence
of {A,] such that 0 < \ A,, /         p k , k = 1, 2, . . . .
                                                                Finally, let E be the projec-
tion with range 9,ll determined by the orthonormal set (p,, J and let W be a par-
tially isometric translormation with initial set @ and final set 9,ll.
     Then WT - TW is in 9 Hence WTW* - TWW* = WTW* - TE is in 9
                                   .                                                         .
But then, since (A,,] is in the spectral set of 9 by choice of that subsequence,
TE is in 9. Hence WTW* is in 9, and therefore W*WTW*W = T is in 9 too,
which is a contradiction.
     I t remains to prove the theorem for the case 9 = F Let us suppose that
B A - A B is in F f o r all B in 9 and that A is not in             Then A is of the form
T X I , where T is in S a n d not in 5. Hence there exists an infinite orthonormal
set (cp,) i n 4 s u c h t h a t Tcp, = Xncpn, Xn # O , n = 1 , 2 , ... , A n # Xmif m # n.
Hence, if IT defined by the equations

in the closed linear manifold determined by (p, J , U = I in @ @ Dl, UTU-' -
T is not in F and hence UT - T CT is not either. Therefore the assumption
that A is not in 3 leads to a contradiction and the theorem is proved for 9 = X
                                 3. A metric in 9/3
   We now confine our attention to the case 9 =            beginning with the
definition of a norm in 9 K/
   Throughout the remainder of the paper we employ the notation ( A I for the
bound of the operator A of 9 .
   DEFINITION Let a be a n arbitrary element of 9/T We define I a I,
called the norm o j a, by the equation
                                 la1 = g . l . b . I A I .
                                             A e a

        3.1.          The norm I a I in 9 / S h a s the following properties:
(1)            IaI   2 0, the equality sign holding if and only if a = 0 ;
(2)                           I a + P l S lal+lPI;
(3)                               IaSI 5 I a l l P I ;
(4)                                I = I a l l I;
(5)                               la*l= lal;
(6)                                 111 = 1.
      If no such subsequence exists, T is in yand hence in 9.
                      BOUNDED OPERATORS IN HILBERT SPACE                               855

   The validity of the laws (2) - (5) is an immediate consequence of the defini-
tion of I a I in terms of the norm I A I in 9, and the fact that the latter function
has those properties. The same is true of the assertion I a I 2 0. Moreover,
in view of (2) and (3) we can conclude that the set of elements a of B/Sfor which
I a I = 0 is a two-sided ideal in B/S. Hence, since S i s a prime ideal in 9, we
have either a = 0 when and only when I a I = 0, or a = 0 for all a in @/XZ1
Thus to complete the proof it is necessary only to establish (6). We have then
to show that

We note first

hence we need only show g. 1. b. I I       + T I = 1 for T a 5, T = T*. But, if T
is so restricted, we can find, for any a > 0, a real number X with I X I < a such
that Tq = A 9 for some 9 # 0 in @, by virtue of the spectral properties of T.
Thus / ( I T)9 I / 19 I > (1 - a) and hence I I                +
                                                          T I exceeds 1 - e. But
then, since e i9an arbitrary positive number, we have I I           +
                                                                  T I h 1. Hence,
since 1 I 1 = 1, (6) follows.
   THEOREM 3.2. With I a - /3 I interpreted as the distance between a and 8,
9ISi.s a com.plete linear metric space.
   That 9 / S i s e metric space follows from Theorem 3.1 while its linear properties
are evident. To show that it is complete we must prove that, for every sequence
{ a , ) in 9/3sych that
                                  lim la,,-a,I           =0,

there exists an element a such that
                                   lim ( a , - a ( = 0.

Let ( a , ] be a sequence satisfying the first of these conditions. We choose a
subsequence ( a , , ) such t h ~ t
                                   I ank - an 1 6   2
                                                    11                        for n 2 nk.

We then choose an arbitrary element A,, of a,, and an element CI of a,,              - a,,
such that I CI I 5 4. Setting A,, = 6'1        +
                                         A,, , we have A,, in a,,

   f l For the fact that T i e divieorleee in 3 implies that %/rcontaine no two-sided ideale
except (0) and B/3iteelf. Cf. [Is],pp. 56-57 for a diecwion of ideale in commutative
rings which is readily generalized to cover the c u e in hand.
856                                            J. W. CALKIN

Continuing this process we determine a sequence ( A , , } , with A,,                e a,,   , such

                    I A,,,, - An, I 5              C I An.+,

                                                               - An* I 5    2z.
Hence there exist an element A of 9 such that
                                          lim I A,, - -41 = 0,

and if a is the residue class to which A belongs we have, in consequence,
                    lim    I   Y
                               (,   - (Y. I   =   lim / a,, - a I = 0.
                   n+m                            k+m 

   We note in passing that the space 9 / 3 i s non-separable. For every idempo-
tent except 0 in %/3can be shown to have the norm 1, and if E(X) is the resolu-
tion of the identity in % of a transformation with spectrum the entire interval
0 $ X 5 1, the set r(X) of idempotents in 9/3such that t(X) contains E(X)
has the property that a(X2)           -
                                e(X1) is an idempotent different from zero if 0 5
X2 < X i 5 1.

                                              4. The space 2
  We now propose to realize 9 / 3 a s an algebraic ring of operators in a certain
complex Euclidean space. To define a space I suitable for this purpose, we
make use of a concept of generalized limit introduced by Banach and ~ a z u r . ~ ~
In the interests of greater generality, however, we shall employ a less restrictive
concept of generalized limit than that of these.writers, and we begin with a dis-
cussion of this concept.2a
  We consider a linear functional defined for all bounded sequences {x,) of
real numbers, denoted by Lim x, , which has the following properties:
                                     n -m 

(b)                ~ i xn 2 0, for xn h 0,
                       m                                        n = 1, 29   . ..;

(c>                Lim xn is independent of x for each integer p;

(d)                Lim1=1.

  " [a, p. 34.
  ' 8 The possibility of generalizing the notion of Banach in this way waa pointed out to us
by J. v. Neumann; originally we had employed the Banach limit.

  For subsequent use, we note that (a) and (b) imply
(e)             Lirnx, 2 Limy,,
                n -rw     n 

                                              if      x, 2 y,         for n = 1, 2,   ....
  The reader will observe that the four preceding conditions differ from the four
basic properties of the Banach limit, as given in the reference cited, in the follow-
ing respects: first, we do not require homogeneity; second, and more important,
the Banach limit has the property
                                Lim xn+l = Lim xn
                                        n-m                   n+m   

in place of our (c). Since from (1) one has
                                     Lim x+I
                                          ,,              =   Lirn x,,
                                     nd m                     n-m 

it is clear that (1) implies (c).*'
   We now wish to show that homogeneity is a consequence of conditions (a) - (d)
and that the use of (c) instead of (1) does not affect the other essential properties
of Linl xn .

  To begin we observe that (a) implies
                         Lim rxn = r Lim x, for all rational r.
                         n-a                  n -00   

We now consider an arbitrary bounded sequence (x,),and rat'ional upper and
lower bounds, R and r, respectively, of (x,). Invoking (2), (d), and (e), we
then obtain
                   R = Lim R 2 Lim xn 1 Lim r = r.
                                n -op              n -m               n -a 

Hence, since R is any rational upper bound of (x,), and r any rational lower
bound, we have
                           1. u. b. x, 2 Lim xn 2 g. 1. b. xn

hioreover, if we now invoke (c) in conjunction with (3), it becomes clear a t once
that we must have
                       lim sup xn 2 Lirn xn 2 lim inf x n
                               n+m                 n+m                  n-w 

Thus, we have the important property
(g)                           lim                lim
                     Lim x, = n--00 x,, whenever n-m xn exists.

   1 The argument of Banach, loc. cit., thus serves t o establish the existence of a functional
withthe properties ( a ) - ( d ) . Anelegant andsimpledirectproof of the existence of such func-
tionalshas been obtained by Ulam and Kakutani independently, but has not beer~published.
Moreover, it is not difficult to show t h a t there exist functionals satisfying (a)-(d) but not
(1). This latter fact, however, is not of essential importance in the present paper, but
rather in certain related investigations. Added in proof: Since the completion of this
paper J. v. Neumann has developed a general theory of limits of the sort used here.
His results will appear in a forthcoming number of the ANNALS MATHEMATICS
                                                                     OF                 STUDIES
dealing with the theory of meaeure.
858                                         J. W. CALKIN

Furthermore, since it is now clear that Lim is a bounded additive functional
                                                     n -00
on the space of all bounded sequences, we can conclude that it is homogeneous:
(h)                  Lim ax, = a Lim x,,               for all numbers a.
                     n -roo             n -+a,

  We now extend the notion of generalized limit to bounded sequences of com-
plex numbers in the obvious way; if {x,) is such a sequence we set
                              Lirn xn = Lim %xn
                              n -00         n -roo
                                                        + i Lirn 5 x n .
                                                             n -roo

I t is then readily proved that properties (a), (c), (d), (g), (h), persist in the com-
plex case.
    Hereafter, as occasion requires, we shall refer to the properties of Lim xn
                                                                                n -roo
as given above by letter without other comment.
   We turn now to the construction of our space Q. First, we consider the class
I" of all sequences f j,,) in Hilbert space $jwhich are weakly convergent to zero;
this class is evidently a module when we define addition and scalar multiplica-
tion by the equations

In I", we define ( {fn , { gn] ) by the equation

invoking the boundedness of the sequences if,] and (g,) to assure the bounded-
ness of the sequence of numbers { (jng,)}. Then from the properties of Lirn
                                                                                    n -m

given above and the properties of the inner product in @, we have

Thus ({j,}, {g,)) has all the properties requisite for an inner product in 2"
except that requiring that ( { j ) , { j ) ) = 0 if and only if Ifn) = 0, and it is easy
                               ,       ,
to see that this requirement is not fulfilled, since Q" contains sequences strongly
convergent to zero. However, as A. E. Taylor [la] has pointed out, this require-
ment is not an essential one, since we can regard it not as a postulate but as a
definition of zero and thus of equality. In our case, this means that we must
identify the sequences { jn and ( g,] provided
                                      Lirn I jn- gn /' 0.
                      BOUNDED OPERATORS IN HILBERT SPACE                                 859

In this way, we obtain a class I' of elements f , g, . . . , the quotient group of the
additive group It' by the subgroup of elements If,) such that
                                     Lim / fn
                                                  l2 = 0.
If under this homomorphism j,) +f, ( g n ) + g, we define

Thus we achieve in I' a (possibly incomplete) complex Euclidean space; that is
to say, I' is a module and the function ( j ,g) has the properties

                                   (f,j )   =   0 implies f   =   0.
Since the pertinent facts in this connection are discussed in the paper of Taylor
cited above, it is unnecessary for us to dwell on them here.
   I t now remains for us to consider the space I' with reference to the matters of
completeness and separability. The answers to both questions are provided
through the two simple lemmas which follow.
   LEMMA   4.1. T h e cardinal number o j I' i s not greater t h a n the cardinal number c
of the continuum.
   Let ( p m ) be an arbitrary complete orthonormal set in $3, ( f , ] an arbitrary
element of I". Then, if

                                     fn =   C an,mpm

 (a,,,) is a bounded matrix. Thus the cardinal number of I" does not exceed
the cardinal number of the class of all bounded infinite matrices, and the cardinal
number of the latter class is c. Hence the cardinal number of I' is certainly
less than c.
   LEMMA     4.2. T h e space Sf contains a n orthonormal set w i t h cardinal number C.
   We consider an enumeration (r,) of the positive rational numbers, a complete
orthonormal set ( p n Jin Q,   and the correspondence rn ++ between them. We
denote by ( ( p : J ) the class of all infinite subsequences of ( p , ] , a running over
some set which we leave undesignated. Now let al and az be any two distinct
positive numbers, ( r : ) and ( r x ) infinite subsequences of (r,) convergent to a1
and a2 respectively. Corresponding to ( r k ) and (r:) we have two subsequences
of ( p , ) which belong to ( (p," J ) ; we denote these by (p,"' ] and (p,"' ) , respectively.
Then, for n larger than some integer N, we have pzl # p:' and hence
                                   Lim (w,"l, p,"') = 0.
                                   n 4-
860                                   J. W. CALKIN


for all a. Hence, if Q denotes the element of I' containing ( c p , " ) J it follows that

the set of all (9,)contains an orthonormal set with cardinal number c.
  THEOREM     4.1. T h e space $' i s incomplete.
  Consider an orthonormal set I$,} in Q' with cardinal number a t least c. Then
the set of all
                  a,$, with    2  / a , j 2 < CY has cardinal number a t least 2', and

hence, in view of Lemma 4.1, this set cannot belong to $'.           Therefore I' is in-

   We now denote by I the space obtained by completing 2'; the details of this
construction are described in [5] and in [13], so we need not consider them here.
   THEOREM    -1.2. T h e dimension number of I i s c.
   Since I' is dense in I, the dimension number of 2 cannot exceed C, by Lemma
4.1. Rut by Lemma 4.2, it cannot be less than c.

            6. The algebraic ring     9 K and congruence modulo T i n $3
   We now consider transformations induced in the space $ by means of members
of the ring 9.
   LEMMA     5.1. Let A be a n arbitrary bounded ever ywhere-de$ned transformation
in $3, ( j , ) a n arbitrary sequence of the class I". T h e n ( A f , ) i s in S" a n d
Lim / A j n i2 = 0 if Lim I fn l2 = 0.
nd   m               n-+m
   That {Aj,) is in 9" follows at once from the fact that a bounded transforma-
tion is weakly continuous. And since we have A j n i2 $ ( A / jn 12, n =
1, 2, . . . , it follows from property (e) of Lim that Lim I j, j2 = 0 implies
                                                 n -+m         n-m
                                   Lim / A f n   l2   =   0.
                                   n -m

   THEOREM Let A be a n arbitrary member of 9. T h e n , i f f i s a n arbitrary
element of I', a n d ( j , } belongs to f , we set

where g i s the element of 9' containing { A j , ) . The transformation T1(A) so de-
fined in I' i s a single valued linear bounded transformation w i t h bound not exceeding
 the bound of A in $5. T h u s T1(A) has a unique closed bounded extension T(A)
with d o m a i n I, and the bound of T(A) does not exceed the bound of A in @.
    That T1(A) is single-valued follows a t once from Lemma 5.1, while its linear
character is a consequence of the linearity of A. Since, in addition,

   z6 This simple proof of Theorem 4.1 w m suggested by J . v. Neumann. The theorem can
also be proved directly.
                              EOUSDED     OPERATORS I N HILBERT SPACE                                  861

it is evident that T 1 ( A )is bounded with bound less than or equal to / A 1. Thus
the transformation p l ( A ) = T ( A ) exists and has domain 9 , while its bound is
clearly the same as that of T l ( A ) .
    THEOREM     5.2. T h e class $97 of operators T ( A ) i n 2, defined for all A in 3,
i s a n algebraic ring of operators in the class of all bounded everywhere deJined opera-
tors in $!, and i s a homomorphism of % with respect to the operations           . ,*, a n d     +,
scalar multiplication; that i s ,

    A 1 of these relations are quite obvious, except possibly T(.4*) = T * ( A ) .
To prove this we consider two arbitrary elements f and g of € ' nd sequences
( f , ) and ( g , ) belonging to f and g, respectively. Then
           ( T ( A ) j ,g )   =   Lim ( A f , , g,)   =   Lim     (fn   , A*gn)   =   ( f , T(A*)g).
                                  n -m                    n -+m

Thus T ( A * ) and T * ( A ) coincide on $!',and therefore throughout $!.
   LEMMA      5.2. A necessary and sz~$cient condition that T ( . 4 ) be thc transforma-
tion in i zchich takes cvery element of 9 into zero i s that -4 belong to thc itleal T o j
totally continriolrs operators i n 9. T h u s T ( A ) = T ( B ) if a n d only if -4 i s congru-
ent to R nzorfulo T
   From the homomorphism 9-+ 91,it follo~vs             that the set of all A in 93 such
that T ( . 4 ) = 0 is a two-sided idcal9 in $8. JIoreover, since a totally continuous
transformation A in $8takes weakly convergent sequences into strongly converg-
ent ones, it follows from property (e) of Lim that 9 2 3- Hence, by Theorem
                                                           n -m
1.4, we have either 9 = T o r 9 = 9. But since 9 clearly fails to cont,ain the
identity in 9 , must concl~lde
                    3 we                       that 9 = Tl which establishes the lemma.
   Since it also fvllows im~nediately               from the homonlorphism 9 --+ ' that % 2
is isomorphic to the ring 9/9,here 9 i s the ideal of all A in 9 such that T ( A ) =
                                            w                                    3
0, we can now conclude that '9lis isomorphic to 9/Y. More precisely, we have
    THEOREM          5.2. T h e algebraic ring 9 s isomorphic to a/y i t h respect to the
                                                         i                         w
operations    +,        . , *, a , , a n element T ( A ) of $97 corresponding to the element a of
g/T if and only i '4 belongs to a .
    DEFIKITION.1. I f a i s a n arbitrary element of %/T we d e j n e T ( a ) a s the
d e m e n t of        corresponding to a under the isomorphism of Theorem 5.3.
    Evidently T ( a ) i:; identical with T ( , 4 ) , for all A in a , and u7e shall continue
to use both notations for elements of 5 as occasion requires.
   THEOREM           5.4. A n element T ( a ) of           i s self-adjoint, partially isometric, or
u n i t a r y , respcctiz1ely if and only i f a has that property in the sense of D e j n i t i o n 2.1.
An element T ( a ) of               i s a projection i j and only i j a i s a n idempotent according
to that definition.
   The asscrtion of the theorem concerning self-adjointness is obvious. T Oprove
the other parts of the theorem we note first that since the properties which form
862                                                 J . W. CALKIN

the various criteria of Definition 2.1 are all defined in terms of the operations
and *, it follows from Theorem 5.3 that an element a of 93/3possesses one of
them if and only if T(a) dorq. But for transformations, each of these properties
is characteristic of the (31n+ of transforn~ations question: moreprecisely, an
everywhere defined bounded linear operator T in 5 is a projection if and only if
f = T* = T , ' ~is partially isometric if and only if T*T is a projection,27and thus
is obviously unitary if and only if T*T and TT* are equal to the identity in Q .
Hence the theorem follo~vs.
   The reader will notc that Theorems 2.3-2.7 can be interpreted now to yield
assertions conccrni1:g the homomorphism 93 -+ %; the details here are obvious
and we omit them.
   THEOREM    5.5. Lct CY                            f
                              an arbitrary elemcnt o % / T Then the bound o j the

operator T(cY) ? zs a ; Ln ot1zc.r words, the isomorphism % / T o % zs un isom-
etry. Thus % is closed zn the unzform topology jor operators.
   Evidently the concluding assertion is a consequence of the first one and The-
orem 3.2. Hence we need oniy show / T(a) I = 1 CY 1.
   From the final statement of Theorem 5.1 and the definition of the norm in
%/3,e have a t once

Hence me need only establish

To prove (2)) we first select an arbitrary element A of a, with canonical decom-
position A = W B . FVe then denote by X the lowest upper bound of those points
of the spectrum of B which are either limit points of the spectrum or characteris-
tic values of infinite multiplicity, and by S the set of points p in the spectrum
of B such that p exceeds X. Then S clearly consists entirely of isolated points,
each a characteristic value of finite multiplicity. Furthermore, either S is a
finite sequence ( p , ] or an infinite sequence with X as limit. Hence if !JXn is
the characteristic manifold of B corresponding to pn , n = 1, 2, . . . , and we set
C = B - X I o n m =nx   ,lJ!     C = Oon @      O,C b e l o n g s t o x Thus, if B1 =
B - C, then Al = W B l belongs to a. hloreover, this is evidently the canonical
decomposition of A 1 , so A, I = / B1 I. Hence we have

  S o w let us consider the transformation T ( B l )in ?. FTe distinguish two cases,
according as the sequence {p,) is infinite or finite. If {p,) is infinite, Tm =
   !En a Hilbert space and contains an infinite orthonormal set {p,). Further-

more, Blpn         =   Xpn , n      =     1, 2,   - .- .   Thus, if cp is the element of 9' to which

    26    [17],Theorems 2.35, 2.36. 

    -27   [9], Lemma 4.3.2. 


(cp,]   belongs, we have
                                  Q)               , )
                           (T(B~)Q, = Lim ( B l ~ n~ n = A.

                                (Q, Q) = Lim ( ~ n~, n = 1.

But clearly X = I B1 , and hence we have

   Now suppose {H,) is finite, and let E(X) be the resolution of the identity for
B1 in @.. Then, since X is a limit point of the spectrum of B1 , there exists a
monotone increasing sequence (A,) with limit X, such that E(X,+,) - E(X,) is
different from zero, n = 1, 2, . . . . Hence we can select an orthonormal set
 (v,) with cp, in the range of E(X,+l) - E(Xn), n = 1, 2, . . . . Moreover, for
every n , we have An $ (Blcq,, v,) 5 A,,+, . Hence if Q is the element of I'
containing ( c p , ) , we have (4) in this case also.
   Finally, since B1 = W*A1 we have T(Bl) = T(W*)T(A1) and since T(W*)
is partially isometric by Theorems 5.1 and 2.6, it follows that we have I T(W*)I =
1, and hence that

holds. But T(A1) = T(cY),     and hence from (3), (5) and (6) we have (2) which
completes the proof of t,he theorem.
  We now wish to prove that 6SIZ is not closed in the weak or strong topologies
for operators. The proof reposes on the following lemma concerning monotone
sequences of projections in Em.
  LEMMA   5.3. Let / T(E,,)) be a sequence of projections i n Em such that T(E,,+~)
T(cn), n = 1, 2, . . . . Then, i f lim T(E,) = 0, T(E,) = 0 for all n greater than
                                    n -w

some integer M .
   By Theorem 5.4, each cf the terms o ( a , ) is an idempotent, and consequently
by Theorem 2.4, contains a projection En of 9. The sequence ( E n ) ,      however,
is evidently not necessarily monotone, and our next step is to show that there
exists a monotone non-increasing sequence (F,) of projections in 9such that
En - F, is in 3; n = 1 , 2 , . . . . We begin by setting F1 = El and then, assuming
that F, is determined for all n 5 N, we show that F N + l can be defined.
   We note first that

and hence that FNEN+1FN EN+1 in 3. Consequently, it follows that
( F ~ E ~ + ~ F ,-' FNENflFN in T. Thus, if '3nNis the range of F N , FNEN+lFN
                 )             is
induces in '3nNa self-adjoint transformation E;+, congruent to its square modulo
the class of all totally continuous operators in (mN . Hence, by Theorem 2.4,
there exists in '3nN a projection F:+~ congruent to ~ ; + 1 modulo that class.
Therefore if F N + 1 is the projection in @ which is equal to F:+~ in ( m N , equal to
864                                    J. W . CALKIN

zero in @ 0 O N t hen F N + i - FNEN+iFN is in 3: But then FN+l EN+1 in 3
                      ,                                                      is
and since F N + L is equal to zero in @ 5, O N , we have F N + I 5 F N Thus the se-

quence (F,) with the stated properties exists.
  Now let us suppose that T(a,) is never zero. Then, if lim T ( c ) = 0, T(c,)
must be different from T(en+l) for an infinite number of values of n. Hence
we can select a subsequence (T(ank)]      such that T(ank+,)< T(an,) holds, k = 1,
2, . . . Consequently, we clearly have F,,+, < Fnk k = 1, 2, . . , since
                                                            ,                 .
T(en) = T(Fn).                                                ,
                      Therefore, if %k is the range of Fnk none of the spaces
% O %k+l is empty, and we can select an orthonormai set ( ( p k ) with
    in % O %k+l, k = 1, 2, . . . . Let %Obe the closed linear manifold deter-
mined by ( ( p k ] , FO projection with range %O Then Fo - FoFnkFothe pro-
                       the                                               is
jection with range determined by (cpj), j = 1, 2, . . . k , and hence belongs to 3.
Consequently T(Fo)T(Fnk)T(Fo) T(F0) and T(Fo)is clearly not zero. Hence,
since we obviously have
                               lim T(Fo)T(Fnk)T(Fo) 0

i lim T(en) = 0, it follows that the latter is impossible.       Therefore T(en) = 0
for all n sufficiently large, as we wished to show.
  I t is of some interest to note the following alternative statement of the pre-
ceding lemma; if ( T(en)] is an infinite sequence of orthogonal projections in %,
and i
    f          T(an) is the identity in I, then all but a finite number of terms of the
        n -1
sequence T(cn) are zero.
   THEOREM The algebraic ring 5!ll i s not closed in either the weak or strong
topology for operators.
   We consider a monotone sequence ( T(an)) of projections in %, with T(cn+l) >
T(en), n = 1, 2, . . . . Such a sequence is readily generated by means of a
sequence (En)of projections in $8 such that -En+1- En is a projection with
range a Hilbert space, n = 1, 2, . . ; we have only to choose en as the residue
class of En . The sequence T(cn), being monotone, is convergent in both the
weak and the strong topologies and has a limit which is a projection. Let us
suppose that this limit is in Em and hence that it has the form T(c) where a
is an idempotent in %/3: Then (T(en) - T(a)) is a monotone non-increasing
sequencz of projections in 5!ll with limit zero and thus T(en) = T(c) for all n
greater than some integer N, by Lemma 5.3. This, however, is impossible in
view of the fact that (T(an))was chosen with T(E,,+~) T(cn), n = 1, 2, . . . ;
we must conclude therefore that the sequence ( T(an)) has no limit in 'X.
   We proceed now to examine the relationship between the spectrum of a self-
adjoint member of % and the corresponding self-adjoint transformations in %.
We begin with a formal definition. .
   DEFINITIONLet A be a n arbitrary self-adjoint transJformation in 4. Then
a poinf X of the spectrum of A which i s a limit point of the spectrum or a character-
                        B O U N D ~ DOPERATORS IN HILBERT SPACE                                      865

istic valz~e znjinzte mult~plzcztyi s called a point of condensatzon of the spectrum.28
T h e set of all such points zs called the condensed spectrum of A . T h e complementary
set in the A-plane zs called the augmented resolz~entset of A.
    THEOREM        5.7. Let A be a self-adjoint transformatzon zn @. T h e n a necessary
and suficient condztzon tlzat X belong to the augmented resolvent set of A zs that the
manzfold IM of solutzons of the equatzon A f - Xf = 0 have ajinzte dzmension number
and that in @ @ %', A - X I induce a transjormatzon wzth bounded inverse.
    Let IM be the manifold of zeros of '4, ill the transformation induced in @ @ !l   ,     I l
by A . Then the condition of the theorem may be stated in this way: YJI is
finite-dimensional and X is in the resolvent set of A 1 . But the latter is possible
if and only if X is ,z finite distance from the spclctrum of A l , which is to say
that X is an isolated point of the spectrum of A or belongs to the resolvent set of
A . Thus the theorem follows.
    THEOREM        5.8. Let A be a self-adjoznt transformatzon zn 4. T h p n the resolvent
set of the transformatzon T ( A ) 2n 1\ zs the augnlcntcd resolvent set of A and the
spectrum of T ( A ) zs the condensed spectrum of A . Every point in the spectrum of
T ( A ) i s a characteristzc value wzth multzplzczty c
    Let belong to the augmented resolvent set of A , and let                  be the manifold
of zeros of A - X I , E the projection opt.rator of 6 @ %'.                 Let B be equal in
YJI to zero, and in $5 @ YJI to the inverse of the transformation induced in that
space by A - XI. Then B is in $3 and B ( A - X I ) = E . Thus
 T ( B ) T ( i l - X I ) = T ( E ) . But l ' ( E ) is the identity in ?, since I - E is in 3;
and hence T ( B ) = [ T ( A ) - AT(I)]-'. Therefore X is in the resolvent set of
    S o w suppose X is In the condensed spectrum of A. Then X is either a char-
acteristic value of infinite multiplicity or a limit point of the spectrum of A ,
and in either ease we can select an orthonormal set ( p a ) in $5 such that

                                        lim ( A p n , A p n ) = k 2
                                       n -+a
          lim Apn -
                        b   n   l2   = lim
                                             (1   Apn   1' + k21 pn l2 - 2 k ( A p n , pn))   = 0.

But then, if I,${     is any subsequence of ( p n ] and (I! the element of I' containing
( q n }we have
                                                  T(A)(I!   =   A$.
hforeover, there exist c such subsequences such that any two have only a finite
number of terms in common and consequently c orthogonal elements of 9'
satisfying the preceding equation. Therefore X is a characteristic value of T ( A )
with multiplicity c.

   z8   These are the Haufungspunkte of the spectrum in the Rense of Weyl, [20].
866                                       J. W. CALKIN

   We have now shown that every point of the augmented resolvent set of A is
in the resolvent set of T ( A ) and that every point of the condensed spectrum of
A is in the spectrum of T ( A ) . But since the augmented resolvent set and the
condensed spectrum of A together constitute the entire A-plane, they must be
respectively the ~ n t i r e  resolvent set and the entire spectrum of A.
   I t should be observed that we capnot infer from theorem 5.8 that the sum of
the characteristic manifolds of T ( A ) is I. Whether or not this is so we are
unable to say a t present.
   \Ye now have from Theorem 5.8 the classical theorem of 'Il'eyl [20].
   THEOREM      5.9. Le! A and B br two self-adjoint transformations i n @, such
that A - B i s totally contznkous. T h e n A and B have the same condensed spectrum
and ?he same augmented resolcent set.29
   The theorem is obvious since T ( A ) = T ( B ) .
   We proceed now to determine the resolution of the identity corresponding to
a self-adjoint transformation in 9 R by means of the resolution of the identity
of a corresponding member of 3. If A is self-adjoint in @, E(X) its resolution
of the identity, it is clear that T ( E ( X ) ) as many of the properties of a resolution
of the identity in F. Specifically, it is readily shown that the following asser-
tions hold:
(1) T ( E ( X ) )permutes with T ( A ) ;
(2) T ( E ( X ) ) T ( E ( p )= T ( E ( P ) ) ( E ( X ) ) = T ( E ( F ) ) or P 5 A ;
                             )             T                           f
(3) T ( E( A ) ) = 0 if X is less than the lower bound of T ( A ) , T ( E ( X ) )= 1 if A
exceeds the upper bound of T ( A );
(4) in the range of T ( E ( X ) ) , he upper bound of T ( A ) does not exceed A, and
in the range of 1 - T ( E ( X ) )the lower bound of T ( A ) is not less than X.
In spite of these facts T ( E ( X ) )fails usually to be the resolution of the identity
for T ( A ) . This is readily seen in view of Lemma 5.3, which assures us that in
general we do not have
                                lim T(E(X   +     6) = T ( E ( X ) ) .

   hloreover, if A and B are self-adjoint and congruent modulo E(X) and F(X)
their respective resolutions of the identity, we do not in general have
E(X) - F(X) in 3 Thus for a self-adjoint transformation T ( A ) in '3ll we can
exhibit many monotone families of projections with the properties (1) - (4)
above, none of which is the resolution of the identity of A .
   We can however, derive from any one of these families of projections the
resoluticn ol' the identity belonqing to the member of 9lt in question. The
procedure is described in the following theorem.
   THEOREM     5.10. Let A be a self-at-ljoint transformation in Q,E(X) its resolution
of the identity. Let E(X) be the family of projections i n f! defined by the equation
                         E(A) = lim T(E(X
                                                + e)),      e   > 0.
   ? @ Weyl proves also that if A is an arbitrary bounded self-adjoint transformation, there

exist6 a totally continuous self-adjoint transformation T such that A  +  T has a pure point
spectrum. . I t present we see no direct way t o derive this result from ours. Cf. also [14].
                        DOUXDED OPERATORS I N HILBERT SPACE                           867

Then E(X) is the resolution of the identity in $ of T ( A ) .
    The existence of the limit E(X) follows of course from the monotone charscter
of the family. T ( E ( X ) )
                           which follows in turn from the corresponding property of
E(X). li'e shall no\\- show that E(X) has the following six properties:
( 1 ) E(X) permutes with T ( A ) ;
( 2 ) E(X)E(p) = E(P)E(X)= E(p) for p 5 A ;
(3) E(X) = 0 for X < a ( a the lower bound of T ( A ) ) ,a nd E(X) = 1 for X 2 b
(b the upper bound of T ( A ) )   ;
(4) limE(X e ) = E(X),e > 0 ;

(5) in th'e range of E(X),the upper bound of T ( A ) does not exceed A ;
(6) in the range of 1 - E(X),the lower bound of T ( A ) is not less than X and if
it is equal to 	A it is not attained.
   The validity of (1) follows a t once from the permutability of T ( A )and T ( E ( X ) ) .
   To prove ( 2 ) )we note first that
            T(E(X   + E ) ) T ( E ( I I=) T(E(p))T(E(X+ e ) )
                                       ) 	                      for all e   > 0.
Thus, allowing e to tend to zero, we have


for all p and all   E   >   0.   Consequently, again allowing e to approach zero, we

Moreover, if p 5 X , we have clearly E(p) 5 E(X), and so E(X)E(II) E(p). =
Therefore (2) holds.
   Kow let a be the lower bound of T ( A ) . Then by Theorem 5.8, a is the lower
bound of the condensed spectrum of A and it follows therefore that if X is less
than a, then the range of E(X) must be finite-dimensional. For otherwise the
spectrum of A would have points of condensation less than or equal to X Thus
T ( E ( X ) )= 0 for X < a and therefore E(X) = 0 for X < a.
   On the other hand, let b be the upper bound of the spectrum of T ( A ) . Then
b is the upper bound of the condensed spectrum of A and by a similar argument
I - E(X) has a finite-dimensional range, X >= b. Thus T(E(X                 +
                                                                    e)) = T(I)
for X 2 b and e > 0. But then E(X) is tile identity in $ for X 2 b. Thus we
have (3).
   Xext we note that for el > E > 0, we have

Hence, allowing t and tl both to approach zero, preserving the relations tl > c,
el > 0, we obtain (4).
   Finally, we consider the behavior of T ( A ) in the range of E(X) and its orthog-
onal complement. Since E(X).-lE(X) has upper bound not exceeding A, it
       that thc upper bound of 1'(E(X
follo~vs                                          +
                                      c ) ) T ( . l ) T(E(X           +
                                                            6 ) ) does not exceed

X  + c. Rut for CI-cryf in F, we hnvc

Hrmcc the r1ppcr bound of E(X)?'(.I)E(X) oc- not exceed X and ( 5 ) is established.
   I n cntlrcly similar fa-hion, it c : ~ nt ) shown that in the range of 1 - E(X) the
lo~ier  bound of 7'1'1) i i not lcsi than X. hIol.c~ovt~r, X is the lower bound and
thi. lotrer 1)ound is attainc~cl,n e ha\ t1 for iomc f In 9 ,

               lirn 17'(.1)11 - E(X
                                           + c ) ) f , ( l - E(X + e ) ) f ) = X / j l2
I3ut i 7'(-1) '1     - E(X + c ) ) f , ( 1 - EiX +
                                                 t))f)          i i monotone non-decreasing as e
q>!)roa(ah~s   z('1.0, :ind c'ori~equcxntlyu e have

for horn(. t > 0 Thi\, hen(,\ c,r is impossible. -incaclin the range of 1 - E(X           +  e),
the loner bound of 7'(=1)s not Ics- than A. Hmce, we havil (6).
   'l'he\c> six properties arc sufficient to charac~terizc~ at, the re~olutionof the
identity of T ( A )     One can for t~xarnplc~,  srguc. as folloi\i. On the basis of these
six propertics the approximation theorem (Theorcm 6) of the paper [GI of 1,engyel
and Stone can be proved and from this result it follows that E(X) permutes with
every bounded linear opc,rator defined ovcr 9 which permutes with T ( A ) ( [ G I ,
Theorem 7 ) . But this fact together with properties ( I ) , ( 5 ) , ( 6 )above uniquely
determines E(X) as the resolution of the identity for T(,1) ( [ G I , Theorem 5).
   JVe are now7 in position to establish c e r t a ~ n relationships between the resolu-
tions of the identity belonging to two self-adjoint transformations in @ whose
difference is totally continuous.
   THEORE,M 11. L r t .4 and B be self-adjoznt transjormatzons zn @ such that
-2 - B zs totally continuous. Lrt E(X) and F(X) be the rrsolutzons of the zdentzty
correspotl-ding to A iand B respectively. T h r n , zj p zs zn the a u g u m ~ n t e drrsolvent
s ~ of -4, E ( p ) - F ( p ) zs totally contznuous.
   1,ct E(X) be the resolution of the identity of T ( A j in 9 . Then, since is in
the resolvent set of T ( A ) ,we have for some 6 > 0, E(p - 6 ) = E(p 4- 6). But,
from Theorem 5.10 it is clear that we have

     [6], Theorem 3, for example.
                        EOT'KDED OPERATORS IN HILBERT SPACE                      869

 Thus T(E(p)) = T(F(p)) and hence E(p) - F(p) belongs Oo 3, we wished to
    I t is important to note that the requirement that p be in the augmented
 resolvent set cannot be dropped from Theorem 5.11. Consider, for example,
 a nonnegative definite self-adjoint transfornlation A in @ which is totally
 continuous and has an inverse, and its resolution of the identity E(X). Consider
 also the transformation 0 in @, and its resolution of the identity F(X). Then
 A - 0 is in 3 but E(0) = 0 and F(0) = I .
    THEOREM 5.12.~'Let A and B be self-adjoint transformations i n @ such that
A - B is, totally continuous, and k t E(X) and F(X) be their respective resolutions
of the identity. Then, if p i s less than A, there exists a totally continuous trans-
formation Th,, , i n B such that E(p) T,,h i s a projection satisfying the inequality

If T,,h can be chosen so that the equality sign holds, then the only points Xo on the
interval p < Xo < X which belong to the spectrum of either A or B are characteristic
values of finite multiplicity.
   Conversely, k t A and B be self-adjoint transformations in @ with resolutions of
the identity E(X) and F(X), respectively. Then, if the inequa1it;es T(E(p)) 4
T(F(X)) and T(F(p)) S T(E(X)) hold for all X and p such that p < X, A - B is
totally continuous.
   We prove the converse part first. From the inequalities in question, it fol-
lows at once that
                 lim T(E(X
                              + e)) = lim T(F(X + el),
                                                                   e   > 0,
for all X and thus that T(A) and T(B) have the same resolution of the identity.
Hence T(A) = T(B), or A - B is in 3.
  Now let A - B belong to 3 and let E(X) be the resolution of the identity of
T(A) in I. Then, for p < X, we have

Hence T(F(X))T(E(p))T(F(X)) = T(E(p)) and therefore F(X)E(p)F(X) - E(p)
is in 3. Consequently, invoking Theorem 2.4 with reference to the ring of
bounded everywhere defined operators in the range of F(X), we see that there
exists a totally continuous operator To in 9, with the value zero in the range of
I - F(X), such that
                                 F(X)E(p)F(X)     + To
is a projection satisfying

  " Compare $4 of [20] to which this theorem is closely related.
Thus, if we set

T,,x , is in 3- and E(p) + T,,x is a projection satisfying the inequality of the
   S o w suppose we have

Then T(E(p)) = T(F(A)), and since for all Xo on the interval p   < Xo < X, we have

it follows that E(X) is constant on that interval. Hence every point on the in-
terval is in the resolvent set of T(A) = T(B), and this is equivalent to the
concluding statement of the first paragraph of the theorem.
   COROLLARY. A, B, E(X), a n d F(X) be a s in Theorem 5.12. T h e n , i f
   f: 1 ,

i s totally continuous.
    From the inequality of Theorem 5.12, we have, for p   < A,

Thus F(X)E(p) - E(p)F(X) is totally continuous for p < A, and by symmetry
for p > X also.
   This is not necessarily so, however, for X = p. For consider any two projec-
tions E and F in @ such that % ( E )and % ( F ) and their orthogonal complements
are Hilbert spaces. Let A be a transformation which is equal to zero in %(E)
and which induces in @ Q % ( E ) a nonnegative definite self-adjoint totally con-
tinaous transformation whose inverse exists. Then A is in T and if E(X) is the
resolution of the identity of A, E(0) = E. Similarly, u7e can define a self-
adjoint totally continuous operator B with resolution of the identity F(X) such
that F(0) = F. But then A - B is in 3; while in general E(O)F(O) - F(O)E(O)
is not.
   I t follo~,salso from the preceding example that the hypothesis p < X of
Theorem 5.12 cannot be replaced by the hypothesis p S X.
   We conclude this paper with a few observations concerning congruence
modulo 3 in tB which are not restricted to self-adjoint transformations.
   THEOREM     5.13. Let Al a n d A2 be members of 9 whose difference belongs to 3;
A1 = WIB1, A2 = W2B2 their c a n a i c a l decompositions. T h e n B1 - B2 i s in 3;
and if zero i s in the augmented resolvent set of B, , Vrl - Wz i s in 3:
   Since A:A~ = B: , A ~ = B:~, we have T(A:A~) = T(B:) = T ( B ~ ) . More-
                     BOUNDED OPERATORS IN HILBERT SPACE                             871

                = [           = T(B:). B u t B1 and Bz are nonnegative definite
over, [ T ( B ~ ) ] ~ T ( B ~ ) ] ~
and hence

for all ( f n J in I", by property ( f ) of Lim.   Hence T ( B , ) and T ( B 2 )are both
                                           n +m
nonnegative definite, and since T ( B : ) can have only one nonnegative definite
square root, we have T ( B 1 )= T(B2) which implies that Bl - B2 is in 3.
  Now let

be the canonical decomposition of T ( A , ) . Then B = T ( B 1 )= T ( B 2 ) . &lore-

Hence we can infer t h a t the initial set of the partially isometric transformation
T(W1)     contains the initial set of W (the closure of the range of B ) and that the
two transformations are equal there. Hence T ( W 1 )is identical with W pro-
vided the initial set of T ( W 1 )is the closure of the range of B. Hut, if the
origin is in the augmented resolvent set of Bl , it is in the resolvent set of
T(B1) = B and hence W is unitary. Thus under the hypothesis of the theorem
we have T ( W 1 )= W. Furthermore, under that hypothesis, the origin is also
in the augmented resolvent set of BZ and hence a similar argument yields the
equation T(W2) = W. "hus T(W1) = T ( W 2 )and W 1 - W 2 is in 3 as we
wished to show.
    We wish to emphasize that the hypothesis t h a t zero be in the augmented
resolvent set of B1 cannot be omitted from Theorem 5.13. For example, con-
sider a totally continuous nonnegative definite self-adjoint transformation A
with an inversit. Then if A = WlBl is its canonical decomposition and 0 =
WzB2 is the canonical decomposition of 0 , u7ehave W 1 = I, W 2 = 0 , A - 0 in 7.
    IVe now extend Theorem 5.9 to cover transformations which are not neces-
sarily self-adjoint.
    DEFINITION Let A be a n arbitrary bounded everywhere defined transforma-
tion in @. We define the augmented resolvent set of A a s the sct of points X i n the
complex plane such that the following conditions are satisjied:
( 1 ) the manifold of zeros of A - X I has a finite dimension number;
(2) the range of A - X I i s closed; 

( 3 ) the orthogonal complement of the range of A - XI has a finite dimension 


T h e complement of the augmented resolvent set i s called the condensed spectrum of A. 

   Evidently the augmented resolvent set of A contains the resolvent set. so t h a t
our terminology is justified.
   THEOREM 5.14. Let A a n d C be two bounded everywhere-defined transforma-
872                                 J W, CALKIN

tions in $5 such t h t A - C i s totally continuous. T h e n A and C have the same
augmented resolvent sets.
    I t is sufficient for the proof to show that every point in the augmented re-
solvent set of A is also in the augmented resolvent set of C .
    Let X be a point of the augmented resolvent set of A, % the manifold of zeros
of A - XI. Then on 9 0 tm,A - XJ induces a transformation T with an
inverse whose range is % ( A - XI). hloreover, since % ( A - XI) is closed,
T-' is bounded.
    Now let 2 be the manifold of zeros of C - XI. Then if in is a Hilbert space,
($I        O ) . 2 is a Hilbert space, and T has a contraction T1 with domain
(,@ 0 O ).2 which also has a bounded inverse. But in (@ 0 YX). Y?, we have
A - C = T 1and this contradicts our hypothesis that A - C is in 3. Hence 2
must have a finite dimension number.
    We now observe that when is a point of the augmented resolvent set of A,
5 is a point of the augmented resolvent set of A*. Hence the preceding argu-
ment serves to show that the manifold of zeros of C* - XI has a finite dimension
number. But this manifold is precisely the orthogonal complement of the
closure of %(C - XI), and hence to show that X is in the augmented resolvent
set of C, it remains only for us to prove that %(C - XI) is closed.
    To establish the latter we consider the canonical decompositions
                     A-XI=        WlB1,      C - X I = W2B4.
From Theorem 5.13, it follows that B1 - B is totally continuous. hloreover,
since the range of A - XI is closed, so also is the range of B1 , while the manifold
of zeros of B1is the manifold of zeros of A - XI. But then the origin is in the
augmented resolvent set of B1 and hence in the augmented resolvent set of Bz .
Hence %(B2)s closed which implies that %(C - XI) is closed also as we wished
to show.
    If X is in the augmented resolvent set of A , we denote by RA the transforma-
tion which is equal to zero in @ @ % ( A - XI) and which takes each element j
of % ( A - XI) into that element g in the orthogonal complement of the manifold
of zeros of ( A - XI) which satisfies ( A - XI)g = j. Thus R A ( A - XI) is the
projection with range the orthogonal complement of the manifold of zeros of
A -- XI. We call the family of transformations Rh SO defined the augmented
resolvect of A.
   THEOREM      5.15. Let A and B be two transjormations in @ such that A - B
i s in 3. Let R:" and Ri2' be respectively their augmented r~solvents. T h e n
R:" - R:~' i s in 3 j o r all X jor which these transjormations are defined.
   Consider the transformation T ( A ) = T ( B ) in I. If X is in the augmented
resolvent set of A (or B ) it is in the resolvent set of T ( A ) ,and T(R:") and T(R:")
are both inverses of T ( A ) - X . 1 = T ( B ) - X . 1. But only one such inverse
can exist; hence T(R:") = T ( R ~ 'and R:" - R?' is in 3:

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