Hilbert–Schmidt integral operator

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

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

is called a Hilbert–Schmidt kernel and the associated integral operator Template:Math given by

$(Tf)(x)=\underset{\mathrm{\Omega }}{\int }k(x,y)f(y)dy,f\in L^2(\mathrm{\Omega }),$

is called a Hilbert–Schmidt integral operator.Template:SfnTemplate:Sfn Then 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 operator may be extended to any locally compact Hausdorff spaces. Specifically, let Template:Math be a separable Hilbert space and Template:Math a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel Template:Math on Template:Math can be reinterpreted as demanding Template:Math belong to Template:Math. Then the operator

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

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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