Operator theory Basic notions: bounded and unbounded operators Concept
Hilbert space adjoint
Summary
The adjoint of a bounded operator between Hilbert spaces is the bounded operator characterised by . It extends the adjoint of a linear map between finite-dimensional inner product spaces, and the conjugate transpose of matrices. The adjoint is used to define self-adjoint, normal, unitary and positive operators, and it makes the bounded operators on a Hilbert space a C*-algebra.
Where this leads
45 pages build on this one.
- Spectral theorem for unbounded self-adjoint operators
- Hilbert–Schmidt expansion theorem
- Toeplitz algebra
- Bilinear series of a symmetric kernel
- Calkin algebra
- Spectral theorem for compact self-adjoint operators
- Courant–Fischer characterization of characteristic values
- Ideal of compact operators
- Atkinson's theorem
- Algebra of bounded operators on a Hilbert space
- Von Neumann bicommutant theorem
- Ultraweak topology
- Trace class operator
- Hyponormal operator
- Unitary operator
- Stone's theorem on one-parameter unitary groups
- Weak operator topology
- Lidskii's trace formula for integral operators
- Von Neumann algebra
- Spectrum of the Laplace–Beltrami operator
- Strang splitting
- Lie–Trotter product formula
- Borel functional calculus
- Picard's criterion for first-kind equations
- Nest algebra
- Adjoint of an unbounded operator
- Douglas' lemma
- Real spectrum of self-adjoint operators
- Positive operator
- Mercer's theorem
- Murray–von Neumann classification of factors
- Operator space
- Weyl–von Neumann theorem
- Kaplansky density theorem
- Spectral triple
- Fuglede's theorem
- Singular values of a compact operator
- Strong operator topology
- Schmidt's singular value decomposition of a kernel
- Spectral theorem for bounded normal operators
- Weyl's theorem on compact perturbations
- Cotlar–Stein lemma
- Reflexive operator algebra
- Commutant
- Tikhonov regularization
Further reading
About this page
| Tag | 6HSJBJ8TTQ |
|---|---|
| URI | m2://6HSJBJ8TTQ |
| Kind | concept |
| Writing | W0 stub |
| MSC | 47A05, 47B02 |
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Used by
- Spectral theorem for compact self-adjoint operators
- Algebra of bounded operators on a Hilbert space
- Von Neumann bicommutant theorem
- Trace class operator
- Hyponormal operator
- Unitary operator
- Von Neumann algebra
- Picard's criterion for first-kind equations
- Adjoint of an unbounded operator
- Douglas' lemma
- Real spectrum of self-adjoint operators
- Positive operator
- Weyl–von Neumann theorem
- Kaplansky density theorem
- Fuglede's theorem
- Singular values of a compact operator
- Schmidt's singular value decomposition of a kernel
- Spectral theorem for bounded normal operators
- Weyl's theorem on compact perturbations
- Cotlar–Stein lemma
- Tikhonov regularization