# Hilbert-Schmidt integral operator

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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)