Hilbert–Schmidt integral operator

Template:Short description In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Template:Math in Template:Math, any Template:Math such that

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

is called a Hilbert–Schmidt kernel. The associated integral operator Template:Math given by

$(Tf)(x)={\int }_{\mathrm{\Omega }}k(x,y)f(y)dy$

is called a Hilbert–Schmidt integral operator.Template:SfnTemplate:Sfn Template:Math is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

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

Hilbert–Schmidt integral operators are both continuous and compact.Template:Sfn

The concept of a Hilbert–Schmidt integral operator may be extended to any locally compact Hausdorff space Template:Math equipped with a positive Borel measure. If Template:Math is separable, and Template:Math belongs to Template:Math, then the operator Template:Math defined by

$(Tf)(x)={\int }_{X}k(x,y)f(y)dy$

is compact. If

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

then Template:Math 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.Template:Sfn

See also

• Hilbert–Schmidt operator

References

• Template:Cite book

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