mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in ''n''- dimensional Euclidean space R^''n'', a Hilbert–Schmidt kernel is a function ''k'' : Ω × Ω → C with :

\int_ \int_ ,  k(x, y) , ^ \,dx \, dy < \infty

(that is, the ''L''²(Ω×Ω; C) norm of ''k'' is finite), and the associated Hilbert–Schmidt integral operator is the operator ''K'' : ''L''²(Ω; C) → ''L''²(Ω; C) given by :

(K u) (x) = \int_ k(x, y) u(y) \, dy.

Then ''K'' is a Hilbert–Schmidt operator with Hilbert–Schmidt norm :

\Vert K \Vert_\mathrm = \Vert k \Vert_.

Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). The concept of a Hilbert–Schmidt operator may be extended to any

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

Hausdorff spaces. Specifically, let ''X'' be a locally compact Hausdorff space equipped with a positive

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...

. Suppose further that ''L''²(''X'') is a separable

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

. The above condition on the kernel ''k'' on Rⁿ can be interpreted as demanding ''k'' belong to ''L''²(''X × X''). Then the operator :

(Kf)(x) = \int_ k(x,y)f(y)\,dy

is compact. If :

k(x,y) = \overline

then ''K'' is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples.

References

* (Sections 8.1 and 8.5) * {{DEFAULTSORT:Hilbert-Schmidt integral operator Operator theory

See also

References