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 Ω in n-dimensional Euclidean space Rn, then the square-integrable function k : Ω × Ω → C belonging to L2(Ω×Ω) such that

ΩΩ|k(x,y)|2dxdy<,

is called a Hilbert–Schmidt kernel and the associated integral operator T : L2(Ω) → L2(Ω) given by

(Tf)(x)=Ωk(x,y)f(y)dy,fL2(Ω),

is called a Hilbert–Schmidt integral operator.[1][2] Then T is a Hilbert–Schmidt operator with Hilbert–Schmidt norm

THS=kL2.

Hilbert–Schmidt integral operators are both continuous and compact.[3] The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let L2(X) be a separable Hilbert space and X a locally compact Hausdorff space equipped with a positive Borel measure. The initial condition on the kernel k on Ω ⊆ Rn can be reinterpreted as demanding k belong to L2(X × X). Then the operator

(Tf)(x)=Xk(x,y)f(y)dy,

is compact. If

k(x,y)=k(y,x),

then T 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.[4]

See also

Notes

  1. Simon 1978, p. 14.
  2. Bump 1998, pp. 168.
  3. Renardy & Rogers 2004, pp. 260, 262.
  4. Bump 1998, pp. 168–185.

References

  • Renardy, Michael; Rogers, Robert C. (2004-01-08). An Introduction to Partial Differential Equations. New York Berlin Heidelberg: Springer Science & Business Media. ISBN 0-387-00444-0.
  • Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge University Press. ISBN 0-521-65818-7.
  • Simon, B. (1978). "An Overview of Rigorous Scattering Theory". S2CID 16913591.