Hilbert-Schmidt integral operator

A bounded linear integral operator $ T $ acting from the space $ L _ {2} ( X, \mu ) $ into $ L _ {2} ( x, \mu ) $ and representable in the form

$$ ( Tf ) ( x) = \int\limits _ { X } K ( x, y) f ( y) \mu ( dy),\ \ f \in L _ {2} ( X, \mu ), $$

where $ K ( \cdot , \cdot ) \in L _ {2} ( X \times X, \mu \times \mu ) $ is the kernel of the operator (cf. Kernel of an integral operator, [1]).

D. Hilbert and E. Schmidt in 1907 were the first to study operators of this kind. A Hilbert–Schmidt integral operator is a completely-continuous operator [2]. Its adjoint is also a Hilbert–Schmidt integral operator, with kernel $ \overline{ {K ( y, x ) }}\; $[3]. A Hilbert–Schmidt integral operator is a self-adjoint operator if and only if $ K ( x, y) = \overline{ {K ( y, x) }}\; $ for almost-all $ ( x, y) \in X \times X $( with respect to $ ( \mu \times \mu ) $). For a self-adjoint Hilbert–Schmidt integral operator and for its kernel the following expansions are valid:

$$ \tag{1 } ( Tf ) ( x) = \ \sum _ { n } \lambda _ {n} ( f, \phi _ {n} ) \phi _ {n} ,\ \ f \in L _ {2} ( X, \mu ), $$

$$ \tag{2 } K ( x, y) = \sum _ { n } \lambda _ {n} \phi _ {n} ( x) \phi _ {n} ( y), $$

where $ \{ \phi _ {n} \} $ is the orthonormal system of eigen functions of $ T $ corresponding to the eigen values $ \lambda _ {n} \neq 0 $. The series (1) converges with respect to the norm of $ L _ {2} ( X, \mu ) $, while the series (2) converges with respect to the norm of $ L _ {2} ( X \times X, \mu \times \mu ) $, [4]. Under the conditions of the Mercer theorem the series (2) converges absolutely and uniformly [5].

If

$$ \int\limits _ { X } | K ( x, y) | ^ {2} \mu ( dy) \leq C \ \ \textrm{ for } \textrm{ all } x \in X, $$

then the series (1) converges absolutely and uniformly, [4].

If $ \mu $ is a $ \sigma $- finite measure, then the linear operator

$$ T: L _ {2} ( X, \mu ) \rightarrow L _ {2} ( X, \mu ) $$

is a Hilbert–Schmidt integral operator if and only if there exists a function $ M ( \cdot ) \in L _ {2} ( X, \mu ) $ such that the inequality

$$ | ( Tf ) ( x) | \leq M ( x) \| f \| $$

is valid for almost-all $ x \in X $( with respect to the measure $ \mu $) [7]. Thus, the Hilbert–Schmidt integral operators form a two-sided ideal in the Banach algebra of all bounded linear operators acting from $ L _ {2} ( X, \mu ) $ into $ L _ {2} ( X, \mu ) $.

Hilbert–Schmidt integral operators play an important role in the theory of integral equations and in the theory of boundary value problems [8], [9], because the operators which appear in many problems of mathematical physics are either themselves Hilbert–Schmidt integral operators or else their iteration to a certain order is such an operator. A natural generalization of a Hilbert–Schmidt integral operator is a Hilbert–Schmidt operator.

References

[1]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[2]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1
[3]	M.H. Stone, "Linear transformations in Hilbert space and their applications to analysis" , Amer. Math. Soc. (1932)
[4]	F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[5]	J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1961) (Translated from French)
[6]	L.V. Kantorovich, B.Z. Vulikh, A.G. Pinsker, "Functional analysis in semi-ordered spaces" , Moscow-Leningrad (1950) (In Russian)
[7]	J. Weidmann, "Carleman operatoren" Manuscripta Math. , 2 : 1 (1970) pp. 1–38
[8]	K. Moren, "Methods of Hilbert spaces" , PWN (1967) (Translated from Polish)
[9]	Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a2]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)