Hilbert-Schmidt integral operator

A bounded linear integral operator acting from the space into and representable in the form

where 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 [3]. A Hilbert–Schmidt integral operator is a self-adjoint operator if and only if for almost-all (with respect to ). For a self-adjoint Hilbert–Schmidt integral operator and for its kernel the following expansions are valid:

(1)

(2)

where is the orthonormal system of eigen functions of corresponding to the eigen values . The series (1) converges with respect to the norm of , while the series (2) converges with respect to the norm of , [4]. Under the conditions of the Mercer theorem the series (2) converges absolutely and uniformly [5].

If

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

If is a -finite measure, then the linear operator

is a Hilbert–Schmidt integral operator if and only if there exists a function such that the inequality

is valid for almost-all (with respect to the measure ) [7]. Thus, the Hilbert–Schmidt integral operators form a two-sided ideal in the Banach algebra of all bounded linear operators acting from into .

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)