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, ).
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 . Its adjoint is also a Hilbert–Schmidt integral operator, with kernel . 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:
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 , . Under the conditions of the Mercer theorem the series (2) converges absolutely and uniformly .
then the series (1) converges absolutely and uniformly, .
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 ) . 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 , , 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.
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Hilbert-Schmidt integral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_integral_operator&oldid=12345