Hilbert–Schmidt Integral Operator

In 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 Hausdorff spaces. Specifically, let ''X'' be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that ''L''²(''X'') is a separable space, separable Hilbert space. 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 operator, 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.

See also

* Hilbert–Schmidt operator

References

* (Sections 8.1 and 8.5)

*

{{DEFAULTSORT:Hilbert-Schmidt integral operator
Operator theory