Английская Википедия: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 Шаблон:Math in Шаблон:Math-dimensional Euclidean space Шаблон:Math, then the square-integrable function Шаблон:Math belonging to Шаблон:Math such that

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

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

<math>(Tf) (x) = \int_{\Omega} k(x, y) f(y) \, dy, \quad f \in L^2(\Omega),</math>

is called a Hilbert–Schmidt integral operator.Шаблон:SfnШаблон:Sfn Then Шаблон:Math is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

<math>\Vert T \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}.</math>

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

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

<math>(Tf)(x) = \int_{X} k(x,y)f(y)\,dy,</math>

is compact. If

<math>k(x,y) = \overline{k(y,x)},</math>

then Шаблон: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.Шаблон:Sfn

See also

• Hilbert–Schmidt operator

References

• Шаблон:Cite book

• Шаблон:Cite book
• Шаблон:Cite web