Hilbert–Schmidt integral operator explained

In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain in -dimensional Euclidean space, then the square-integrable function belonging to such that

\int _{\Omega} \int _{\Omega} | k(x, y) | ^{2} dx dy < inf ty , is called a Hilbert–Schmidt kernel and the associated integral operator given by(Tf) (x) = \int _{\Omega} k(x, y) f(y) dy, f \in L ^{2(\Omega),} is called a Hilbert–Schmidt integral operator.Then is a Hilbert–Schmidt operator with Hilbert–Schmidt norm\Vert T \Vert _{HS} = \Vert k \Vert _{L ^{2}} .

Hilbert–Schmidt integral operators are both continuous and compact.

The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let be a separable Hilbert space and a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel on can be reinterpreted as demanding belong to . Then the operator

(Tf)(x) = \int _{X} k(x,y)f(y)dy, is compact. If k(x,y) = \overline{k(y,x)}, then 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 also

• Hilbert–Schmidt operator

References

• Book: Renardy, Michael . Rogers . Robert C. . An Introduction to Partial Differential Equations . Springer Science & Business Media . New York Berlin Heidelberg . 2004-01-08 . 0-387-00444-0.
• Book: Bump, Daniel . Automorphic Forms and Representations . Cambridge University Press . 1998 . 0-521-65818-7.
• Web site: Simon . B. . An Overview of Rigorous Scattering Theory . 1978 .