Hilbert-Schmidt integral operator
A bounded linear integral operator acting from the space into and representable in the form
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://www.encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_integral_operator&oldid=22571