Hilbert-Schmidt integral operator

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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 Rn, a Hilbert-Schmidt kernel is a function k : Ω × Ω → C with

\int_{\Omega} \int_{\Omega} | k(x, y) |^{2} \,dx \, dy < + \infty

and the associated Hilbert-Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by

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

Hilbert-Schmidt integral operators are both continuous (and hence bounded) and compact.

[edit] See also

[edit] References

  • Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 356. ISBN 0-387-00444-0.  (Sections 7.1 and 7.5)