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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

$\underset{\mathrm{\Omega }}{\int }\underset{\mathrm{\Omega }}{\int }|k(x,y){|}^{2}dxdy<\mathrm{\infty }$

(that is, the L²(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L²(Ω; C) → L²(Ω; C) given by

$(Ku)(x)=\underset{\mathrm{\Omega }}{\int }k(x,y)u(y)dy.$

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

$\Vert K{\Vert }_{\mathrm{H}\mathrm{S}}=\Vert k{\Vert }_{L^2}.$

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 L²(X) is a separable Hilbert space. The above condition on the kernel k on Rⁿ can be interpreted as demanding k belong to L²(X × X). Then the operator

$(Kf)(x)=\underset{X}{\int }k(x,y)f(y)dy$

is compact. If

$k(x,y)=\overline{k(y,x)}$

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.

See also

• Hilbert–Schmidt operator

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534