Hilbert–Schmidt integral operator

In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain $Ω$ in $n$ -dimensional Euclidean space $R n$ , then the square-integrable function $k : Ω \times Ω \to C$ belonging to $L 2 (Ω\timesΩ)$ such that

\int _{\Omega }\int _{\Omega }|k(x,y)|^{2}\,dx\,dy<\infty ,

is called a Hilbert–Schmidt kernel and the associated integral operator $T : L 2 (Ω) \to L 2 (Ω)$ given by

(Tf)(x)=\int _{\Omega }k(x,y)f(y)\,dy,\quad f\in L^{2}(\Omega ),

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

\Vert T\Vert _{\mathrm {HS} }=\Vert k\Vert _{L^{2}}.

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 $L 2 (X)$ 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 $Ω \subseteq R n$ can be reinterpreted as demanding $k$ belong to $L 2 (X \times X)$ . Then the operator

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

is compact. If

k(x,y)={\overline {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]

Notes

^ Simon 1978, p. 14.
^ Bump 1998, pp. 168.
^ Renardy & Rogers 2004, pp. 260, 262.
^ 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".

[FOOTNOTESimon197814-1] Simon 1978, p. 14.

[FOOTNOTEBump1998168-2] Bump 1998, pp. 168.

[FOOTNOTERenardyRogers2004260,_262-3] Renardy & Rogers 2004, pp. 260, 262.

[FOOTNOTEBump1998168–185-4] Bump 1998, pp. 168–185.

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v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
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Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
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Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

See also

Notes

References