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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
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
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Received September 15, 2013. UGUR GÖNÜLLÜ
Department of Mathematics and Computer Science Istanbul Kültür University Bakirköy, 34156, istanbul, TURKEY E-mail: u.gonullu@iku.edu.tr
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.
