Научная статья на тему 'Trace class and Lidskii trace formula on Kaplansky—Hilbert modules'

Trace class and Lidskii trace formula on Kaplansky—Hilbert modules Текст научной статьи по специальности «Математика»

CC BY
73
4
i Надоели баннеры? Вы всегда можете отключить рекламу.
Область наук
Ключевые слова
KAPLANSKY-HILBERT MODULE / CYCLICALLY COMPACT OPERATOR / GLOBAL EIGENVALUE / TRACE CLASS / LIDSKII TRACE FORMULA

Аннотация научной статьи по математике, автор научной работы — Gönüllü Uğur

In this paper, we introduce and study the concepts of the trace class operators and global eigenvalue of continuous $\Lambda$-linear operators in Kaplansky—Hilbert modules. In particular, we give a variant of Lidskii trace formula for cyclically compact operators in Kaplansky—Hilbert modules.

i Надоели баннеры? Вы всегда можете отключить рекламу.
iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.
i Надоели баннеры? Вы всегда можете отключить рекламу.

Текст научной работы на тему «Trace class and Lidskii trace formula on Kaplansky—Hilbert modules»

Владикавказский математический журнал 2014, Том 16, Выпуск 2, С. 29-37

УДК 517.98

TRACE CLASS AND LIDSKIï TRACE FORMULA ON KAPLANSKY-HILBERT MODULES

U. Gôniillii

In this paper, we introduce and study the concepts of the trace class operators and global eigenvalue of

continuous Л-linear operators in Kaplansky-Hilbert modules. In particular, we give a variant of Lidskiï

trace formula for cyclically compact operators in Kaplansky-Hilbert modules.

Mathematics Subject Classification (2000): 47B60, 46A19, 46B99, 46L08, 47B07.

Key words: Kaplansky-Hilbert module, cyclically compact operator, global eigenvalue, trace class,

Lidskiï trace formula.

1. Introduction

Kaplansky-Hilbert module or AW*-module arose naturally in Kaplansky's study of AW*-algebras of type I [2]. I. Kaplansky proved some deep and elegant results for such structures, and therefore they have many properties of Hilbert spaces. In [7] A. G. Kusraev established functional representations of Kaplansky-Hilbert modules and AW*-algebras of type I by spaces of continuous vector-functions and strongly continuous operator-functions, respectively. The functional representations are the main technical tool used in this paper. Cyclically compact sets and operators in lattice-normed spaces were introduced by A. G. Kusraev in [5] and [6], respectively. In [8] (see also [9]) a general form of cyclically compact operators in Kaplansky-Hilbert modules, which, like the Schmidt representation of compact operators in Hilbert spaces, as well as a variant of the Fredholm alternative for cyclically compact operators, was also given. Recently, cyclically compact sets and operators in Banach-Kantorovich spaces over a ring of measurable functions were investigated in [1, 3, 4].

In this paper, we introduce and study the concepts of the trace class operators and global eigenvalue and multiplicity of a global eigenvalue, and give a variant of Lidskiï trace formula for cyclically compact operators in Kaplansky-Hilbert modules. We refer to [9] for the whole standard terminology and detailed information.

2. Preliminaries

A C * -module over the Stone algebra Л is a Л-module X equipped with a Л- valued inner product (■ | •) : X x X ^ Л satisfying the following conditions:

(1) (x | x) ^ 0; (x | x) =0 ^ x = 0;

(2) (x | y) = (y | x)*;

(3) (ax + by | z) = a (x | z) + b (y | z);

© 2014 Gôniillii U.

(4) X is complete with respect to the norm |||ir||| := ||(ir | for all x, y, z in X and a, b in A. As well as its scalar-valued norm |||-|||, a C*-module X has a vector norm, given by |ir| := \J (x \ x). It is not difficult to deduce |||ir||| = ¡¡|ic||| and the Cauchy-Bunyakovskii-Schwarz inequality | (x \ y) | ^ MM-

A Kaplansky-Hilbert module or an AW*-module over A is a unitary C*-module over A that enjoys the following two properties:

(1) let x be an arbitrary element in X, and let (e^be a partition of unity in P(A) with e^x = 0 for all £ G S; then x = 0;

(2) let (x^)^es be a norm-bounded family in X, and let (e^)^es be a partition of unity in P(A); then there exists an element x G X such that e^x = e^x^ for all £ G S

where P(A) denotes complete Boolean algebra of all projections p of A (i. e., p2 = p and p* = p). We say that X is faithful if for every a G A the condition ax = 0 for all x G X implies that a = 0.

Throughout this paper the letters X and Y denote faithful Kaplansky-Hilbert modules over A. Moreover, Q and H will denote an extremally disconnected compact space and a Hilbert space, respectively.

Let Ba(X, Y) denote the set of all continuous A-linear operators from X into Y. In case X = Y, Ba(X) := Ba(X, X) is an AW*-algebra of type I with center isomorphic to A [2, Theorem 7]. Every continuous A-linear operator is dominated and bo-continuous [9, Theorem 7.5.7.(1)]. Furthermore, for every continuous A-linear operator T,

|T|1 = sup {|Tir| : x £ X, \x\ < l} = sup {|Tir| : x £ X, \x\ = l} ,

holds, and |T| e Orth(A) [9, Theorem 5.1.8.], whence we can identify |T|l and |T| since Orth(A) = A.

Let B be a complete Boolean algebra. Denote by PrtN(B) the set of sequences v : N ^ B which are partitions of unity in B. For vi,v2 G PrtN(B), the symbol vi ^ v2 abbreviates the following assertion: if m, n G N and v1(m) A v2 (n) = 0B then m < n. Given a mix-complete subset K C X, a sequence s : N ^ K, and a partition v G PrtN(B), put sv := mixneN v(n)s(n). A cyclic subsequence of s : N ^ K is any sequence of the form (sVk)kgN, where (vk)keN C PrtN(B) and vk <c vk+1 for all k G N. A subset C C X is said to be cyclically compact if C is mix-complete and every sequence in C has a cyclic subsequence that converges (in norm) to some element of C. A subset in X is called relatively cyclically compact if it is contained in a cyclically compact set. An operator T G Ba(X, Y) is called cyclically compact if the image T (C) of any bounded subset C C X is relatively cyclically compact in Y. The set of all cyclically compact operators is denoted by K(X, Y).

Let x G X, y G Y. Define the operator dx,y : X ^ Y by the formula

0x,y(z) := (z | x) y, z G X,

and note that 9x,y G K(X, Y).

The techniques employed in [1] yield the following theorem: U = Su is a cyclically compact opeartor on C# (Q,H) if and only if there is a comeager set Qo in Q such that u(q) is a compact operator on H for all q G Q0.

3. The Trace Class

In this section, we study the trace class operators on Kaplansky-Hilbert modules and investigate the dualities of the trace class.

From now onward, it will be assumed that (ek)keN, (/k)keN, and (rfk)keN verify the representation of a cyclically compact operator T as in [9, Theorem 8.5.6]

3.1. Definition. Let 1 ^ p < <. The symbol Sp(X, Y) denotes the set of all cyclically

j.

compact operators T such that (/x^fceN is o-summable in A. Put vp(T) :=

S1(X,Y) and S2(X, Y) are called the trace class and the Hilbert-Schmidt class, respectively.

3.2. Proposition. Let T G K(X, Y). Then T is in Sl(X, Y) if and only if there exist families (xi)i&i in X and (?/»)«=/ in Y such that (|#i||yi|)-e/ is o-summable and T = bo-E i&idxi,yi- In particular, if (Xi)i€i and (yi)iei are projection orthonormal families and (di)i&i is a family with positive elements, then v\(T) = OLi\xi\\yi\.

< If T is in Si (X, Y), then the result follows from xn := rfnen and yn := /n.

For the converse, assume that the families (xi)ie/ and (yi)ie/ satisfy the stated conditions. The inequality

k k k / \ = E (Ten | /n) = ^ i o-£ (en | Xi) {y» | /n) I

n=1 n=1 n=1 \ is/ /

is/ \\n=i J \n= i J J is/

holds for each k G N, and the proof is finished. >

3.3. Corollary. Let T G Si(X,F) and A G A. Then Vl(XT) = |A|t>i(T) and |T| ^ vi(T) and

Vi (T) = inf i o-J2\x,]\y,] : {Xi)i&1 C X, (yi)ieI C Y I ^ iei >

where (xi)ie/ and (yi)ie/ satisfy condition (ii) of Proposition 3.2.

3.4. Lemma. Let T G S1(X). Then the net (|(Te | e)|)eeE is o-summable in A for all

projection bases E, and the sum o-YÏ,eeE (Te | e) is the same for all projection bases E of X.

< It is enough to observe that there exist a positive cyclically compact operator R1 and a cyclically compact operator R2 in S2(X) such that T = R1R2 and (Te | e) = (R2e | R1e) hold for every e g E, namely,

R1 := bo-Yl ,fk , R2 := bo-Yl rf/2 °ek ,fk . >

k= k=

The trace of T g Sl (X) is defined by tr(T) := o-^eeE (Te | e) where E is a projection bases of X. Observe that vL (T) = tr(T) is satisfied for every positive operator T in Sl (X) and tr(T) = o-£ie/ (y» | x») where (x»)ie/ and (yi)ie/ satisfy the condition (ii) of Proposition 3.2, and so tr is a A-linear operator.

3.5. Lemma. The following statements hold:

(i) tr : (S1 (X),vL(■)) ^ A is a dominated and bo-continuous A-linear operator. In particular, |tr(T)| ^ vi(T) and |tr| = 1;

(ii) tr (T*) = tr (T)* (T G Si (X));

(iii) tr(TL) = tr(LT) whenever TL, LT g Sl(X) (T g K(X) and L g BA(X));

(iv) If Te Si (Y,X) and L G BA(X,Y), then TL G Si(X), LT G Si (F) and |tr(TL)| ^ vi(T)\L\.

< (i) Using the representation of T, we deduce |tr(T)| ^ vi(T). Thus, tr is bo-continuous and subdominated, and hence it is dominated, by virtue of [9, Theorem 4.1.11.(1)].

(ii) Follows immediately from the definition of tr.

(iii) Use the representation of T to obtain tr(LT) = tr(TL).

(iv) If (xj)jS/ and (yi)i£j satisfy the condition (ii) of Proposition 3.2 for T, then (L*Xj)ie/ and (yi)i£j also satisfy the same conditions for TL. Therefore, we have TL G S1(X) and the inequality

ie/

|tr(TL)|= = (LVi \Xi) ^o-^lL^II^KlLlo-^l^ll^;

ie/ ie/ ie/

and so the desired inequality follows from Corollary 3.2. >

Let be a Banach-Kantorovich space. Denote by the set of all A-linear

operators r/ : —> A such that (3 c e A) \rj(x)\ ^ c\x\ (Vx £ ¿2T), and note that consists of all |||-|||-continuous A-linear operators n : X ^ A.

3.6. Theorem. If < : S1(Y,X) ^ K(X, Y)* is defined by <(T)(A) = tr(TA) for all

A £ K (X, Y) and T £ S1 (Y, X), then < satisfies the following properties:

(i) < is a bijective A-linear operator from S1(Y, X) to K(X, Y)*;

(ii) vl(T) = \<p(T)\ (T£^(Y,X)).

< By Lemma 3.5 (i) and (iv), < is a well-defined dominated A-linear operator, and \<p(T)\ < vi(T) holds for all T £ (Y,X). Let <j> £ JfT{X,Y)*. Since J^2(X,F) is a Kaplansky-Hilbert module, s2(X,Y) is in S2(X, Y)* and there exists a unique S £ S2(X, Y) such that 0|S2(XjY) = (-,S). Thus, 0|S2(X,Y)(A) = tr(S*A) (A £ S2(X,Y)). Assume that (xk)fegN, (Vk)feen, and (Ak)keN satisfy representation of S* as in [9, Theorem 8.5.6]. Define Pm '■= i ^fc.^fc (m e N), and note that |PTO| ^ 1. Thus, the following inequality

m

\<f>\ = № > MpJ > №Pm)\ = Ms*pm)\ =

k=1

implies that S* £ S1(Y, X). Because <(S*) is bo-continuous, <(S*)(A) = 0(A) is satisfied for all A £ Jf(X,Y). Thus, tp is onto and ^(S1*)! ^ v\(S*) holds, and the proof is complete. >

The proof of the following lemma can be extracted from the proof of [10, Proposition 1.3].

3.7. Lemma. If the mapping a : X x Y ^ A satisfies the properties:

(i) a(Ax1 + ^X2,V) = Aa(x1,v) + ^a(x2, y) (x1,X2 £ X, y £ Y, A,^ £ A);

(ii) <t(x, Xyi +/«/2) = X*<t(x, yi) + /j,*<j(x, y2) (x £ X, yi,y2 £Y, £ A);

(iii) There exists some A e A+ such that \a(x,y)\ ^ A|ir||y| (a; £ X, y £Y) then there exists a unique A £ B\(X, Y) such that ^ A and a(x, y) = (Ax \ y).

3.8. Theorem. Ifip : (BA(X,Y), | |) ->■ (.^(Y, X)*,¡-1) is defined by 4>(L)(T) = tr (TL) for all L £ Ba(X, Y) and T £ S1 (Y, X), then ^ satisfies the following properties:

(i) ^ is a bijective A-linear operator from Ba(X, Y) to S1(Y, X)*;

(ii) |L| = |^)li (L£BA(X,Y)).

< By Lemma 3.5 (i) and (iv), ^ is a well-defined dominated A-linear operator, and |V>(£)li < 1^1 holds for all L £ BA(X,Y). Let r G J^i(Y,X)*. Define a : X x Y ->■ A by a(x, y) := t(0y,x), and observe that

|<7(a:,j/)| = |r(^;;c)| < Irl^i^) < HMM-

Therefore, there exists A £ B\(X, Y) with a(x, y) = {Ax | y). This implies that ^(A)(9y,x) = T{QytX) and \Ax\2 < |n|Ac||a;|. Thus, we have \A\ < ^ and ip(A)(T) = r(T) (T e Si(Y, X)), and the proof is finished. >

4. Lidskii trace formula

Our main aim in this section is to prove the Lidskii trace formula for cyclically compact operators in Kaplansky-Hilbert modules.

Set [A] = inf {n £ P(A) : nA = A}, the support of A in A.

4.1. Definition. Let T be an operator on X. A scalar A £ A is said to be an eigenvalue if there exists nonzero x £ X such that Tx = Ax. A nonzero eigenvalue A is called a global eigenvalue if for every nonzero projection n £ A with n ^ [A] there exists a nonzero x £ nX such that Tx = Ax.

4.2. Proposition. Let T be a continuous A-linear operator on X and A be a nonzero

scalar. Then the following statements are equivalent:

(1) The scalar A £ A is a global eigenvalue of T;

(2) There is x £ X such that Tx = Xx and |ir| £ *p(A) with |ir| ^ [A].

< (2) ^ (1) : Obvious.

(1) ^ (2) : Let A be a global eigenvalue of T. Consider the set

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

C := {(M,ir) : \x\ £ qj(A), 0 < |ir| < [A] , Tx = Xx} .

The definition of global eigenvalue and [2, Lemma 4.] yield [A] = sup {|ir| : (7r, x) £ c}. From this and the Exhaustion Principle, there exists an antichain (ya)ae^ in P(A) such that suPaeA^a = M> and f°r each a e ^ there is (\xa\,xa) £ C with ¡j,a ^ |ira|. Hence, we get x := bo-J2aeA l^aXa with |ir| = [A] and Tx = Xx, whence the proof. > Let T be in BA(X, Y). For an eigenvalue A of T define

NX := U ker(T - AI)n.

n€ N

The following lemma gives a relation between N\ and ker(T — AI)n (n £ N)

4.3. Lemma. Let T be a cyclically compact operator on X and A be a global eigenvalue of T. If n is a nonzero projection with n ^ [A], then there exist a nonzero projection j with j ^ n and n £ N such that jNx = j ker(T — AI)n.

< Assume by way of contradiction that the assertion is false. Then a sequence (xn)neN can be constructed such that xn £ 7r( (ker(T — XI)n)± n ker(T — XI)n+1) and 7r = \xn\. Therefore, it follows from

(T — AI)n ((T — AI)xn — Axm — (T — AI)xm) = 0 (m < n)

that (T — AI)xn — Axm — (T — AI)xm £ ker(T — AI)n, and so

|2 I , , ,,m \ t\ \ im \ t\ m2

|Txn - TxmY = \Xxn + ((T - XI)xn - Xxm - (T - XI)xm)Y

-X

>\X\2\xn\=7t\X\2 ¿0

> \^xnf + |(T - AI)xn - Xxm - (T - XI)xm\2

which contradicts cyclically compactness of T. This proves the lemma. >

Let T be a cyclically compact operator on X. For a global eigenvalue A of T and for each N £ N define

Pn(A) := sup {n £ P(A) : nNA = n ker(T - A/)N, n ^ [A]} .

Using the lemma above, we immediately have the following corollary.

4.4. Corollary. Let T be a cyclically compact operator on X and A be a global eigenvalue

of T. The following conditions are satisfied:

(1) Pn(A) ^ Pn+1(A);

(2) pn (A)Na = pn (A) ker(T - A/)N;

(3) [A] =sup{pN(A) : N £ N}.

According to [9, Theorem 7.4.7(2)], for each N £ N, there exists a partition (6^)^eS of pN (A) such that NA is a strictly )-homogeneous Kaplansky-Hilbert module over A. Since T is cyclically compact, ) must be a finite number. From [9, Theorem 7.4.7.(1)], we can assume that 2 = N and k(ta,n(n)) = n where ta,n(n) := 6n. So, there is a unique sequence (ta;1)1€N in P(A)N such that ta>1 := (ta>1 (n))neN is a partition of p^(A) and ta,i(n)NA = TA;1(n) ker(T — A/)' is a strictly n-homogeneous Kaplansky-Hilbert module over T\,i(n)A. Moreover, TA;1(n) ^ TA;1+1(n) and TA;1(n) A tA,k(m) = 0 are satisfied for all k,l,m, n £ N with n = m. So, (TA(n))neN is a partition of [A] where tA(n) := sup1eN {TA;1(n)}.

Now, we define the multiplicity of global eigenvalues of cyclically compact operators on X which is an element of the universally complete vector lattice (ReA) which in turn is the universal completion of ReA.

4.5. Definition. Let T be a cyclically compact operator on X and A be a global

eigenvalue of T. The multiplicity of A will be denoted by rA and is described as follows:

ta := o-TjnTA(n) = o-\Jnsup{TA1(n)} = sup {nTA1(n)} £ (ReA)^.

neN neN 1eN 1'™eN

Now, we define the multiplicity of global eigenvalues of cyclically compact operators on X which is an element of the universally complete vector lattice (ReA)which in turn is the universal completion of Re A.

4.6. Lemma. Let U = S^ be in End (C# (Q, H)) and A be a global eigenvalue of U. Then

there is a meager subset such that A(q) is a nonzero eigenvalue of u(q) for all q £ Aa \ B0.

< By Proposition 4.2, Ux = Xx is satisfied for some x G (Q,H) with |ir| = [A]. Thus, u(q)x(q) = A(q)x(q) holds for all q £ Q0 := domu H domx. Define B0 := Qg u (Aa \ {q £ Q : A(q) = 0}), and note that B0 is a meager set in Q. The lemma follows. >

4.7. Lemma. Let U = Su be a cyclically compact operator on C# (Q,H) and A be

a global eigenvalue of U. Then there is a meager subset A0 such that for all q £ AA \ A0 the following equality holds:

ker(U — A/)(q) := (ker(U — A/)) (q) = ker(u(q) — A(q)/).

< Clearly, q £ domu implies ker(U — A/)(q) C ker(u(q) — A(q)/). As U is a cyclically compact operator, there exists a partition of [A], (6k)keN in P(A) such that 6nker(U — A/) is a strictly n-homogeneous Kaplansky-Hilbert module over 6nC(Q). Fix k £ N. Let {ej : i = 1,...,k} be a basis for 6kker(U — A/). Then for some meager set Ak the set {ej(q) : i = 1,..., k} is a basis of ker(U — A/)(q) for all q £ Vk \ Ak, where Vk is the clopen

set corresponding to the projection 6k. From the lemma above we obtain a meager subset B0 such that A(q) is a nonzero eigenvalue of u(q) for all q £ Aa \ B0. Define

Ck := {q £ Vk \ (Ak U Bg) : ker(U — A/)(q) = ker(u(q) — A(q)/)}.

Then we can see that Ck is meager, and so A0 = (Aa \ (|JkeN Vk)) U ((JkeN Ak U Ck) U B0 is meager. Therefore, ker(U — A/)(q) = ker(u(q) — A(q)/) holds for all q £ Aa \ A0, as desired. >

An immediate consequence of the preceding results is the following.

4.8. Corollary. Let U = S^ be a cyclically compact operator on C# (Q,H) and A be

a global eigenvalue of U. Then there exists a meager set B0 such that for all q £ Aa \ B0 the following statements hold:

(1) A(q) is a nonzero eigenvalue of compact operator u(q);

(2) (ker(U — A/)ke (q) = ker(u(q) — A(q)/)k (k £ N);

(3) NA(q) = NA(q) where NA(q) is the generalized eigenspace, corresponding to the eigenvalue A(q) ;

(4) T\(q) = m(\(q)) where m(\(q)) is the algebraic multiplicity of A(q).

Denote by Sp*(u(q)) the set of all non-zero eigenvalues of u(q), that is Sp*(u(q)) = Sp(u(q)) \ {0}.

4.9. Lemma. Let U = S^ be a cyclically compact operator on C# (Q, H) and let £ be a finite subset of C(Q) consisting of global eigenvalues of U and the set

AM C {q £ dom(u) : Sp*(u(q)) \ {a(q) : a £ £} = 0}

be non meager in Q. If Aq is in Sp*(u(q)) \ {a(q) : a £ £} for each q £ AM, then there is a global eigenvalue A of U and a comeager set Q0 that satisfy the following conditions:

(1) [A] = VNeN nN where nN is the projection corresponding to clopen set UN := int(cl(AN)) with

An := {q £ AM : (Va £ £)|a(q) — Aq| ^ 1/N and |Aq| ^ 1/N} ;

(2) 7Tn\M ^ jjirN and irN\cr - A| ^ jn^n (N £ N, a £ £);

(3) If q is in An n Q0, then |A(g)| ^ and \<j(q) — A(g)| ^ jjv hold for each a £

(4) If A(q) = 0 holds for some q £ Q0, then A(q) £ Sp*(u(q)) \ {a(q) : a £ £};

(5) If A(q) = 0 holds for some q £ Q0, then q / Au.

< Without loss of generality we may assume that u(q) is a compact operator on H for each q £ domu. Since AM = |JNeN AN is not meager, UNo = 0 holds for some N0. Let hq be an eigenvector of u(q) corresponding to Aq with ||hq|| = 1 for every q £ Au. For every N, n £ N and q £ UN H AN we can find a clopen set Uq>n)N C UN such that

\u(w)hq - Xqhq\\ ^ i and |cr(w) - Aq| ^ ^ (cr G S)

for all w G Uq;„)N n dom u. We can establish a global eigenvalue AN of U such that [AN] = and

|Aw| ^ — 7rw and 7TAf|cr - AAf| ^ —7rW (cr G S).

Therefore, if we define A := ni Ai + o^ N eN(nN+i — )An+i, then [A] = V N eN and A is a global eigenvalue of U. This and Proposition 4.6 complete the proof. >

4.10. Theorem. Let T be a cyclically compact operator on X. Then there exists

a sequence (Ak)keN consisting of global eigenvalues of T or zeros in A with the following properties:

(1) |Afe| < \T\, [Afe] ^ [Afe+i] (k g N) and o-lim Ak = 0;

(2) There exists a projection in A such that n^|Ak| is a weak order-unity in n^A for all k £ N;

(3) There exists a partition (nk) of the projection such that n0A 1 =0, nk ^ [Ak], and nkAk+m = 0, m,k £ N;

(4) nAk+m = nAk for every nonzero projection n ^ + nk and for all m,k £ N;

(5) Every global eigenvalue A of T is of the form A = mixkeN (pkAk), where (pk)keN is a partition of [A].

< The theorem will be proved in case of X = C# (Q, H) and T = Su General case can be obtained by the functional representations of Kaplansky-Hilbert modules and bounded linear operators on them (see [9, Theorems 7.4.12 and 7.5.12]). Now, by induction and Lemma 4.9, a sequence (An) consisting of global eigenvalues of SU or zeros, and a decreasing sequence of comeager sets (Qn), can be established as follows:

(i) if An(q) = 0 holds for some q £ Qn, then An(q) £ Sp*(u(q)) \ {A^q) : i = 1,... ,n — 1};

(ii) if An(q) = 0 holds for some q £ Qn, then Sp*(u(q)) \ {A^(q) : i = 1,... ,n — 1} = 0;

(iii) Sp*(u(q)) = {An(q) : An(q) = 0 (n £ N)} is satisfied for all q £ Q0 := f| Qn.

Define := /\keN [Ak] and no := [Aiand nk := [Ak] A [Ak+i]X (k £ N). Then this implies (2), (3) and (4). Moreover, since |An(q)| ^ ||u(q)|| and limk^^ Ak(q) = 0 hold for all q e Qo, we have |A„| ^ |f/| and o-lim Ak = 0, and so (1) follows. Let A be a global eigenvalue of U. Then we can assume that the meager set A0 satisfies the condition of the Lemma 4.6. From (iii) we have (A^ n Q0) \ A0 = UkeN Ak where

Ak := {q £ Aa \ Ao : A(q) = Ak(q)} (k £ N).

Since Ak \ int(cl(Ak)) is nowhere dense, [A] = VkeN jk and jkA = jkAk where jk denotes the projection corresponding to the clopen set int(cl(Ak)). Thus, there exists a partition (pk)keN of [A] such that A = mixkeN pkAk holds, and the proof is finished. >

Let (Ak)fceN be as in Theorem 4.10. If Ak = 0 for some k e N, take T\k = 0.

4.11. Definition. The sequence (Ak(T))keN, where Ak(T) := Ak is given by the above

theorem, is called a global eigenvalue sequence of T with the multiplicity sequence (rk(T))k€N where rk(T) := T\k.

4.12. Theorem (Lidskii trace formula). Let T be in S1 (X) and (Ak(T))keN be a global

eigenvalue sequence of T with the multiplicity sequence (rk(T))ke^. Then the following equality holds

tr(T) = o-5>fe(T)Afe(T).

keN

< As in Theorem 4.10, the theorem will be proved in case of X = C# (Q,H) and T = Su Let (Ak (T))keN be a global eigenvalue sequence of T with the multiplicity sequence (~Tk(T))kGn- From Corollary 4.8 and Theorem 4.10, there exists a comeager set Qo such that for each q £ Q0 the following statements hold:

(i) tr(T)(q) = tr(u(q)) and Vi(T)(q) = vi(u(q));

(ii) Sp*(u(q)) = {An(T)(q) : An(T)(q) = 0};

(iii) An(T)(q) = Am(T)(q) if An(T)(q) = 0 or Am(T)(q) = 0 for n = m;

(iv) if Ak(T)(q) ± 0, thenrfc(T)(<?) = m(\k(T)(q)) G N where m(Xk(T)(q)) is the algebraic multiplicity of Ak(T)(q).

From (i), (ii), (iii), (iv) and Lidskii trace formula for the compact operator u(q), we see that

trCT)(q) = tr (u(q)) = ^ fk (T) (q)\k (T) (q)

keN

is absolutely convergent on the comeager set Q 0, and so we have

tr(T) = o-J2rk(T)\k(T). >

keN

References

1. Ganiev I. G., Kudaybergenov K. K. Measurable bundles of compact operators // Methods Funct. Anal. Topology.-2001.-Vol. 7, № 4.-P. 1-5.

2. Kaplansky I. Modules over operator algebras // Amer. J. Math.—1953.—Vol. 75, № 4.—P. 839-858.

3. Kudaybergenov K. K., Ganiev I. G. Measurable bundles of compact sets // Uzbek. Mat. Zh.—1999.— № 6.-P. 37-44—[in Russian],

4. Kudaybergenov K. K. V-Fredholm operators in Banach-Kantorovich spaces // Methods Funct. Anal. Topology.-2006.-Vol. 12, № 3.-P. 234-242.

5. Kusraev A. G. Boolean valued analysis of duality between universally complete modules // Dokl. Akad. Nauk SSSR.-1982.-Vol. 267, № 5.-P. 1049-1052.

iНе можете найти то, что вам нужно? Попробуйте сервис подбора литературы.

6. Kusraev A. G. Vector Duality and Its Applications.—Novosibirsk: Nauka, 1985.—[in Russian].

7. Kusraev A. G. On functional representation of type I AW*-algebras // Sibirsk. Math. Zh.—1991.— Vol. 32, № 3.-P. 78-88.

8. Kusraev A. G. Cyclically Compact Operators in Banach Spaces // Vladikavkaz Math. J.—2000.—Vol. 2, № l.-P. 10-23. "

9. Kusraev A. G. Dominated Operators.—Dordrecht etc.: Kluwer Academic Publishers, 2000.

10. Wright J. D. M. A spectral theorem for normal operators on a Kaplansky-Hilbert module // Proc. London Math. Soc.-1969.-Vol. 19, № 3.-P. 258-268.

Received September 15, 2013. UGUR GÖNÜLLÜ

Department of Mathematics and Computer Science Istanbul Kültür University Bakirköy, 34156, istanbul, TURKEY E-mail: [email protected]

КЛАСС ОПЕРАТОРОВ СО СЛЕДОМ И ФОРМУЛА ЛИДСКОГО В МОДУЛЯХ КАПЛАНСКОГО - ГИЛЬБЕРТА

Гёнюллю У.

Вводятся и изучаются класс операторов со следом и глобальные собственные значения непрерывных гомоморфизмов в модулях Капланского — Гильберта. В частности, устанавливается вариант формулы Лидского о следе для циклически компактных операторов в модулях Капланского — Гильберта.

Ключевые слова: модуль Капланского — Гильберта, циклически компактный оператор, глобальное собственное значение, класс операторов со следом, формула Лидского о следе.

i Надоели баннеры? Вы всегда можете отключить рекламу.