Operator theory Compact and Fredholm operators Concept
Compact operator
Summary
A compact operator is a linear map between Banach spaces that sends bounded sets to relatively compact sets. Compact operators are the operators closest to finite-dimensional linear algebra: their nonzero spectrum consists of eigenvalues of finite multiplicity, which can accumulate only at . They include the finite-rank operators and their norm limits, and integral operators with continuous kernels. They are the setting of the Fredholm alternative and of the spectral theorem for compact self-adjoint operators.
Where this leads
51 pages build on this one.
- Collectively compact operator approximation
- Hilbert–Schmidt expansion theorem
- Nuclear operator
- Toeplitz algebra
- Céa's lemma
- Bilinear series of a symmetric kernel
- Calkin algebra
- Vanishing mean oscillation
- Fredholm property of elliptic operators
- Bifurcation from a characteristic value
- Lax–Milgram theorem
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Minimum total potential energy principle
- Ideal of compact operators
- Atkinson's theorem
- Fredholm index
- Existence of an eigenvalue for a nonzero symmetric kernel
- Boundary integral method for the Dirichlet problem
- Schatten class
- Strictly singular operator
- Lidskii's trace formula for integral operators
- Fredholm map
- Spectrum of the Laplace–Beltrami operator
- Convergence of approximations for second-kind equations
- Schauder's theorem on adjoints of compact operators
- Lomonosov's invariant subspace theorem
- Spectral theory of compact operators
- Poincaré inequality
- Ill-posedness of first-kind equations
- Compactness of integral operators
- Picard's criterion for first-kind equations
- Galerkin method
- Essential spectrum
- Jentzsch's theorem on positive kernels
- Discrete spectrum
- Mercer's theorem
- Fredholm operator
- Lidskii's theorem
- Weyl–von Neumann theorem
- Ladyzhenskaya–Babuška–Brezzi condition
- Prolate spheroidal wave functions
- Projection method for integral equations
- Spectral triple
- Singular values of a compact operator
- Krein–Rutman theorem
- Schmidt's singular value decomposition of a kernel
- Regularization of ill-posed problems
- Weyl's theorem on compact perturbations
- Fredholm's theorem
- Rellich–Kondrachov theorem
Further reading
About this page
| Tag | M2NEP1K5AE |
|---|---|
| URI | m2://M2NEP1K5AE |
| Kind | concept |
| Writing | W0 stub |
| MSC | 47B07 |
| Specializes | Continuous linear map |
| Same as |
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What links here
Used by
- Collectively compact operator approximation
- Nuclear operator
- Toeplitz algebra
- Calkin algebra
- Fredholm property of elliptic operators
- Bifurcation from a characteristic value
- Spectral theorem for compact self-adjoint operators
- Ideal of compact operators
- Atkinson's theorem
- Fredholm index
- Existence of an eigenvalue for a nonzero symmetric kernel
- Boundary integral method for the Dirichlet problem
- Schatten class
- Strictly singular operator
- Convergence of approximations for second-kind equations
- Schauder's theorem on adjoints of compact operators
- Lomonosov's invariant subspace theorem
- Spectral theory of compact operators
- Ill-posedness of first-kind equations
- Compactness of integral operators
- Picard's criterion for first-kind equations
- Essential spectrum
- Discrete spectrum
- Fredholm operator
- Lidskii's theorem
- Weyl–von Neumann theorem
- Prolate spheroidal wave functions
- Projection method for integral equations
- Spectral triple
- Singular values of a compact operator
- Krein–Rutman theorem
- Regularization of ill-posed problems
- Weyl's theorem on compact perturbations
- Fredholm's theorem
- Rellich–Kondrachov theorem
Definitions
Results about this
- Spectral theorem for compact self-adjoint operators
- Schauder's theorem on adjoints of compact operators
- Lomonosov's invariant subspace theorem
- Spectral theory of compact operators
- Ill-posedness of first-kind equations
- Compactness of integral operators
- Weyl–von Neumann theorem
- Krein–Rutman theorem