Abstract
Full Text
UDC 517.43
MATHEMATICS
V. B. KOROTKOV
ON INTEGRAL OPERATORS
(Presented by Academician S. L. Sobolev on 20 V 1970)
1°. Let (E) be a real (B)-space, and let (S(a,b)) be the space of measurable functions. In 1938 L. V. Kantorovich and B. Z. Vulikh ((^1)) introduced and studied a class of operators acting from (E) into (S(a,b)) and having the form
[
(T(x))(s)=(x,\varphi(s)),\quad x\in E,
\tag{1}
]
where (\varphi(s)) is a function with values in the conjugate space (E^).
For separable or reflexive (B)-spaces (E), L. V. Kantorovich and B. Z. Vulikh showed that an operator (T:E\to S(a,b)) has the form (1) if and only if (T) is a (bo)-linear operator ((^2)).
Operators of the form (1) are a generalization of integral operators: if (E) and (E^) are spaces of measurable functions and ((x,x^)=\int x(t)x^*(t)\,dt), then (1) has the form
[
(Tx)(s)=\int K(s,t)x(t)\,dt,\quad x\in E,
\tag{2}
]
where the kernel (K(s,t)=\varphi(s)) is measurable in (t) for almost all (s\in(a,b)) and satisfies the condition (K(s,\cdot)\in E^*,\ s\in(a,b)).
It is known ((^{1-3})) that, under additional restrictions on (E) and on the operator (T), the kernel in (2) is jointly measurable in the variables; in ((^1,^2)) it was shown that if the operator (T) is (bo)-linear from (L_p(a,b)) into (L_q(a,b)), then the kernel in (2) is jointly measurable in the variables and, moreover, satisfies the condition (||K(s,t)|_{p'}|_q<\infty,\ 1/p+1/p'=1).
In ((^3)) this result was generalized to the case of (bo)-linear operators from (X(0,1)) into (Y(0,1)), where (X(0,1),Y(0,1)) are spaces of measurable functions belonging respectively to the classes (P) and (P\cup Q), introduced by D. A. Vladimirov ((^3)).
It should be noted that the proofs of the propositions cited from ((^{1-3})) rely essentially on the separability of the space from which the operator acts.
The difficulties that arise in representing the function (\varphi(s)) in the form of a jointly measurable kernel, and the restrictions on spaces and operators associated with them, are due to the fact that the function (\varphi(s)) in (1) is weakly measurable. These difficulties largely disappear if the function (\varphi(s)) is strongly measurable ((^4,^5)).
In this connection there arises the problem of conditions for representability of an operator in the form (1) with a strongly measurable function (\varphi(s)). Below a solution of this problem is given for operators acting from an arbitrary normed space (real or complex) into the space (S(X,\mu)), where ((X,\mu)) is an arbitrary space with a (\sigma)-finite measure. Further, a broad class of spaces (F(Y,\nu)) of (\nu)-measurable functions is indicated (((Y,\nu)) is an arbitrary space with a (\sigma)-finite measure) possessing the property that strongly measurable functions (\varphi(s)) with values in these spaces generate, by the equality (\varphi(s)=\overline{K(s,\cdot)}), kernels measurable jointly in the variables, and propositions are given on
* From (1) it follows that the function (\varphi(s)) is weakly measurable.
integral representability of linear operators acting from normed spaces of measurable functions into spaces of measurable functions.
2°. Let (E) be a normed space (real or complex), (L) a linear manifold in (E), and ((X,\Sigma,\mu)) a space with a (\sigma)-finite measure. A linear operator (T:L\to S(X,\mu)) will be called a (C)-operator if
[
(Tx)(s)=(x,\varphi(s)),\quad x\in L,
]
where (\varphi(s):X\to E^{*}) is a strongly (\mu)-measurable function, which we shall call the kernel of the operator (T). In what follows we shall assume that the closure of (L) has a topological complement in (E) ((^{6})).
We shall call a set (H\subset L^{\infty}(X,\mu)) (\sigma)-weakly compact if there exists an at most countable set of disjoint sets (X_n,\ n\in I), such that (\mu(X_n)<\infty), (\mu\bigl(X\setminus\bigcup_{n\in I}X_n\bigr)=0), and the sets (P_n(H),\ n\in I), are weakly compact in (L^\infty(X,\mu)), where
[
P_nf=\chi_{X_n}f,
]
(\chi_{X_n}) is the characteristic function of the set (X_n).
A linear bounded operator (Q:L\to L^\infty(X,\mu)) will be called (\sigma)-weakly completely continuous if, for every bounded set (G\subset L), the set (Q(G)) is (\sigma)-weakly compact in (L^\infty(X,\mu)).
Theorem 1. A linear operator (T:L\to S(X,\mu)) is a (C)-operator if and only if there exists a function (\Lambda\in S(X,\mu)), (\Lambda>0), such that the operator
[
(\tau x)(s)=(Tx)(s)/\Lambda(s),\quad x\in L,
]
is a (\sigma)-weakly completely continuous operator from (L) into (L^\infty(X,\mu)).
Remark. If (L) is everywhere dense in (E), then the kernel of the operator (T) is determined uniquely up to (\mu)-equivalence.
Corollary. Let (E=H), where (H) is a Hilbert space. An operator (T:H\to L_2(X,\mu)) is a Hilbert–Schmidt operator ((^{7})) if and only if the operator (T) has a majorant ((^{8})) belonging to (L_2(X,\mu)).
In the case when (H) and (L_2(X,\mu)) are separable, the corollary coincides with Proposition 2.10 ((^{9})).
Theorem 2. Let (E) be reflexive, and let (T:L\to S(X,\mu)) be a linear operator. The following assertions are equivalent: 1) (T) has an abstract norm ((^{2})); 2) (T) has a majorant ((^{8})); 3) (T) is a (C)-operator; 4) (T) has the form
[
(Tx)(s)=(x,\varphi(s)),
]
where (\varphi) is weakly (\mu)-measurable; (T) is a (b_0)-linear operator, i.e., it maps null sequences from (L) into sequences converging to zero (\mu)-almost everywhere.
3°. Let ((Y,\Xi,\nu)) be a space with a (\sigma)-finite measure, and let (F(Y,\nu)) be a normed space of (\nu)-measurable functions*. We shall say that (F(Y,\nu)) is (\sigma)-embedded in (L_1(Y,\nu)) if there exists an at most countable set of disjoint sets (Y_n,\ n\in I), such that (\mu(Y_n)<\infty),
[
\mu\Bigl(Y\setminus\bigcup_{n\in I}Y_n\Bigr)=0
]
and
[
|P_nf|{L_1(Y,\nu)}\le c_n|f|,
]
where (P_nf=\chi_{Y_n}f) and (c_n) does not depend on (f).
Lemma. Let (\varphi(s):X\to F(Y,\nu)) be a strongly (\mu)-measurable function and let (F(Y,\nu)) be (\sigma)-embedded in (L_1(Y,\nu)). Then there exists a ((\mu\times\nu))-measurable function (K(s,t)) such that (\varphi(s)=K(s,\cdot)).
Theorem 3. Let (G(Y,\nu)), ([G(Y,\nu)]^{}) be spaces of (\nu)-measurable functions and
[
(x,x^{})=\int_Y x(t)\overline{x^{}(t)}\,d\nu(t),\quad x\in G(Y,\nu),\ x^{}\in [G(Y,\nu)]^{},
]
([G(Y,\nu)]^{}) is (\sigma)-embedded in (L_1(Y,\nu)), and (L) is a linear manifold in (G(Y,\nu)). If the operator (T:L\to S(X,\mu)) is a (C)-operator, then the operator (T) is an integral operator with a ((\mu\times\nu))-measurable kernel (K(s,t)), satisfying the condition
[
|K(s,\cdot)|_{[G(Y,\nu)]^{*}}\in S(X,\mu).
]
* By a space (Z(Y,\nu)) of measurable functions, here and below in the article we mean a space of classes (f) of (\nu)-equivalent functions satisfying the conditions: a) if (f\in Z(Y,\nu)), then also (|f|\in Z(Y,\nu)), and (|f|_Z=||f||_Z); b) if (f\in Z) and (|g|\le |f|), (g\in S(Y,\nu)), then (g\in Z) and (|g|_Z\le |f|_Z).
4°. Theorem 4. Let 1) (H(Y,\nu)), (G(Y,\nu)), (F(X,\mu)) be normed spaces of measurable functions; 2) (L=H(Y,\nu)\cap G(Y,\nu)) be everywhere dense in (H(Y,\nu)) and (G(Y,\nu)); 3) (G(Y,\nu)) satisfy the conditions of the preceding Theorem 3 and be reflexive; 4) (H(Y,\nu)) be a Banach space; 5) (T:H(Y,\nu)\to F(X,\mu)) be a regular operator. In order that the operator (T) be an integral operator with a ((\mu\times\nu))-measurable kernel satisfying the condition (|K(s,\cdot)|{[G(Y,\nu)]^{*}}\in S(X,\mu)), it is necessary and sufficient that there exist a function (\Lambda\in S(X,\mu)) such that, for all (f\in L), (|(Tf)(s)|\leq \Lambda(s)|f|) for (\mu)-almost all (s\in X).
5°. The results obtained in ((^{8,9})) for operators in separable spaces can be extended to the nonseparable case. Thus, assertions 1), 3) of Theorems 1–5 ((^{8})) remain valid also in the case when (\mu,\mu_{0}) are not separable (*) (\sigma)-finite measures. Assertions 2) of Theorems 1–5 ((^{8})) are formulated taking into account the separability of the range of operators of type ((SC)). We give, for example, analogues of assertions 2) of Theorems 1, 2 ((^{8})).
Theorem 5. Let the measure (\mu) be (\sigma)-finite and not purely atomic. An operator (T) is an operator of type ((SC)) if and only if the adjoint operator (T^{}) is densely defined, the residual spectrum of the operator (T^{}) contains (0), and the range of the closure of the operator (T) is separable.
Theorem 6. Let the measure (\mu) be (\sigma)-finite and not purely atomic. An operator (T) is an operator of type ((C)) if and only if the adjoint operator (T^{}) is densely defined and there exists a symmetric operator (A) such that (A\subseteq T^{}), the residual spectrum of the operator (A) contains (0), and the range of the adjoint operator (A^{*}) is separable.
6°. Theorem 7. An operator (T:L_{2}(Y,\nu)\to L_{2}(X,\mu)) is an integral operator with a ((\mu\times\nu))-measurable Hilbert—Schmidt kernel if and only if there exists a majorant ((^{8})) of the operator (T) belonging to (L_{2}(X,\mu)).
In the case (X=Y=(a,b)), (\mu=\nu) is Lebesgue measure, the theorem follows from a theorem of L. V. Kantorovich and B. Z. Vulikh (((^{2})), p. 332; see also ((^{3})), p. 773).
Corollary 1. Let (A:L_{2}(Z,\xi)\to L_{2}(Y,\nu)) and (B:L_{2}(X,\mu)\to L_{2}(W,\eta)) be linear bounded operators, and let (T:L_{2}(Y,\nu)\to L_{2}(X,\mu)) be an integral operator with a ((\mu\times\nu))-measurable Hilbert—Schmidt kernel. Then (BTA) is an integral operator with an ((\eta\times\xi))-measurable Hilbert—Schmidt kernel.
Corollary 2. An operator (T:L_{2}(Y,\nu)\to L_{2}(X,\mu)) is a Hilbert—Schmidt operator ((^{7})) if and only if the operator (T) is an integral operator with a ((\mu\times\nu))-measurable Hilbert—Schmidt kernel.
In the separable case the assertion of the corollary is well known (((^{4})), p. 102).
7°. The results of the present article can be extended to the case where (X) is a separable locally compact space and (\mu) is a Radon measure.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk
Received
11 V 1970
REFERENCES
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- L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional analysis in partially ordered spaces, M.—L., 1950.
- D. A. Vladimirov, Siberian Math. J., 8, 4, 764 (1967).
- N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, “Nauka,” 1966.
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- V. B. Korotkov, DAN, 190, No. 6, 1274 (1970).
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* That is, (L_{2}(X,\mu)), (L_{2}(X_{0},\mu_{0})) are not separable.