Operator theory
421 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 166 of its 172 ideas have a page so far.
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- ConceptOperator theory
Operator theory is the part of functional analysis that studies linear operators between normed spaces, above all bounded operators on Banach and Hilbert spaces. It asks how far the theory of matrices, with eigenvalues, adjoints and diagonalisation, extends to infinite dimensions. Its tools are used for differential and integral equations, quantum mechanics and signal processing.
- ConceptEndomorphism
An endomorphism of a vector space is a linear map from the space to itself.
- ConceptContinuous linear map
A continuous linear map between normed spaces is a linear map that is continuous for the norm topologies. Such maps are the same as the bounded linear maps, which multiply lengths by at most a fixed constant.
- ConceptOperator norm
The operator norm of a continuous linear map measures the largest factor by which the map can stretch a vector. It makes the space of continuous linear maps into a normed space.
- ConceptInvertible operator
A bounded operator between normed spaces is invertible when it has a bounded inverse. Invertibility is the basic notion behind the spectrum: a number lies in the spectrum of exactly when is not invertible. In infinite dimensions, an injective operator need not be surjective, and a bijective operator needs a theorem to have a bounded inverse.
- TheoremNeumann series
Let be a Banach space, and let be a continuous linear map with operator norm . Write for the identity map of and . Then the following hold.
Map of the subject
Basic notions: bounded and unbounded operators 17 of 18
- Operator theory concept
- Endomorphism definition
- Continuous linear map definition
- Operator norm definition
- Invertible operator definition
- Neumann series statement
- Exponential of a bounded operator definition
- Unbounded operator definition
- Densely defined operator definition
- Graph of an operator definition
- Closed operator definition
- Closable operator definition
- Closure of an operator definition
- Hellinger–Toeplitz theorem statement
- Dual operator definition
- Hilbert space adjoint definition
- Adjoint of an unbounded operator definition
- Cayley transform definition
Spectral theory: foundations 17 of 17
- Spectral theory concept
- Spectrum of an operator definition
- Resolvent set definition
- Resolvent definition
- Eigenvalue definition
- Eigenspace definition
- Point spectrum definition
- Continuous and residual spectrum definition
- Approximate point spectrum definition
- Essential spectrum definition
- Discrete spectrum definition
- Spectral radius definition
- Spectral radius formula statement
- Riesz projector definition
- Holomorphic functional calculus concept
- Numerical range definition
- Toeplitz–Hausdorff theorem statement
Invariant subspaces and dilations 9 of 10
- Invariant subspace definition
- Invariant subspace problem problem
- Lomonosov's invariant subspace theorem statement
- Beurling's theorem statement
- Sz.-Nagy's dilation theorem statement
- von Neumann's inequality statement
- Commutant lifting theorem statement
- Nevanlinna–Pick interpolation statement
- Sylvester equation problem
- Douglas' lemma statement
Particular families of operators 25 of 25
- Self-adjoint operator definition
- Real spectrum of self-adjoint operators statement
- Symmetric operator definition
- Normal operator definition
- Unitary operator definition
- Isometry definition
- Partial isometry definition
- Shift operator example
- Positive operator definition
- Loewner order definition
- Operator monotone function definition
- Square root of a positive operator definition
- Polar decomposition statement
- Orthogonal projection definition
- Fuglede's theorem statement
- Subnormal operator definition
- Hyponormal operator definition
- Nilpotent operator definition
- Multiplication operator example
- Composition operator definition
- Toeplitz operator definition
- Hankel operator definition
- Volterra operator example
- Krein–Rutman theorem statement
- Bishop–Phelps theorem statement
Compact and Fredholm operators 20 of 20
- Compact operator definition
- Ideal of compact operators concept
- Finite-rank operator definition
- Schauder's theorem on adjoints of compact operators statement
- Spectral theory of compact operators statement
- Fredholm alternative statement
- Strictly singular operator definition
- Fredholm operator definition
- Fredholm index definition
- Atkinson's theorem statement
- Calkin algebra object
- Weyl's theorem on compact perturbations statement
- Weyl–von Neumann theorem statement
- Hilbert–Schmidt operator definition
- Trace class operator definition
- Schatten class definition
- Nuclear operator definition
- Lidskii's theorem statement
- Singular values of a compact operator definition
- Fredholm determinant definition
Spectral theorems and functional calculus 10 of 10
- Spectral theorem statement
- Spectral theorem for compact self-adjoint operators statement
- Spectral theorem for bounded normal operators statement
- Spectral theorem for unbounded self-adjoint operators statement
- Projection-valued measure definition
- Continuous functional calculus concept
- Borel functional calculus concept
- Rayleigh quotient definition
- Min-max theorem statement
- Hilbert–Pólya conjecture conjecture
Self-adjoint extensions and perturbation theory 5 of 5
- Essentially self-adjoint operator definition
- Deficiency indices definition
- Von Neumann's theorem on self-adjoint extensions statement
- Friedrichs extension definition
- Perturbation theory of linear operators concept
Operator semigroups and groups 9 of 9
- C0-semigroup definition
- Generator of a semigroup definition
- Hille–Yosida theorem statement
- Dissipative operator definition
- Lumer–Phillips theorem statement
- Contraction semigroup definition
- Analytic semigroup definition
- Stone's theorem on one-parameter unitary groups statement
- Lie–Trotter product formula statement
Ordinary and partial differential operators 13 of 14
- Sturm–Liouville theory concept
- Laplacian definition
- Schrödinger operator definition
- Elliptic operator definition
- Gårding's inequality statement
- Weyl's law statement
- Hearing the shape of a drum problem
- Dirac operator definition
- Atiyah–Singer index theorem statement
- Floquet theory concept
- Almost Mathieu operator example
- Ten martini problem problem
- Anderson localization concept
- Lieb–Thirring inequality statement
Integral and pseudodifferential operators 10 of 12
- Integral operator definition
- Hilbert–Schmidt integral operator definition
- Schur test statement
- Fredholm integral equation problem
- Volterra integral equation problem
- Singular integral operator definition
- Hilbert transform example
- Calderón–Zygmund lemma statement
- Fourier multiplier definition
- Wiener–Hopf method concept
- Pseudo-differential operator definition
- Fourier integral operator definition
Nonlinear operators and fixed points 13 of 14
- Contraction mapping definition
- Banach fixed point theorem statement
- Nonexpansive map definition
- Brouwer fixed-point theorem statement
- Schauder fixed-point theorem statement
- Schaefer's fixed point theorem statement
- Kakutani fixed-point theorem statement
- Markov–Kakutani fixed-point theorem statement
- Ryll-Nardzewski fixed-point theorem statement
- Caristi fixed-point theorem statement
- Leray–Schauder degree definition
- Monotone operator definition
- Browder–Minty theorem statement
- Coercive bilinear form definition
Equations involving nonlinear operators 7 of 7
- Variational inequality problem
- Ekeland's variational principle statement
- Palais–Smale compactness condition definition
- Mountain pass theorem statement
- Bifurcation theory concept
- Newton–Kantorovich theorem statement
- Proximal operator definition
Operator spaces and operator algebras 11 of 11
- Algebra of bounded operators on a Hilbert space object
- Weak operator topology definition
- Strong operator topology definition
- Operator space definition
- Operator system definition
- Completely positive map definition
- Stinespring dilation theorem statement
- Nest algebra definition
- Reflexive operator algebra definition
- Toeplitz algebra object
- Von Neumann algebra definition
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Concepts 63
- Fredholm theory
- Set-valued analysis
- Crossed product
- Fixed-point property of posets
- Inverse problem
- Krein space
- Least fixed point
- Pincherle derivative
- Positive element
- Beurling–Ahlfors transform
- Calderón–Zygmund operator
- Characteristic value of a kernel
- Determinant line bundle
- Eigenfunction of an integral operator
- Elliptic partial differential equation
- Index of a singular integral operator
- Index theory
- Matrix function
- Nemytskii operator
- Positive linear operator
- Positive solutions of integral equations
- Riesz potential
- Spectrum of an element
- Trace of an integral operator
- Abstract Volterra equation
- Adjoint of a linear map
- Analytic index
- Collectively compact operator approximation
- Commutant
- Infinitesimal generator of a Markov process
- Projection method for integral equations
- Rayleigh quotient for symmetric kernels
- Smale's degree for proper Fredholm maps
- Square-integrable kernel
- Tikhonov regularization
- Weak-type estimate
- Wiener–Hopf equation
- Critical point theory
- Differential equations in Banach spaces
- Frame operator
- Koopman operator
- Pseudospectrum
- Regularization of ill-posed problems
- Restriction of an endomorphism to an invariant subspace
- Symbol class
- Bifurcation from a characteristic value
- Differential operator between vector bundles
- Pick matrix
- Riesz potential operator
- Ill-posedness of real Laplace inversion
- Principal symbol of an operator on a manifold
- Ultraweak topology
- Elliptic operator on a compact manifold
- Leray–Schauder method
- Eigenvalue perturbation
- Principal symbol
- Bessel potential
- Fractional Laplacian
- Mild solution
- Lyapunov–Schmidt reduction
- Fixed-point iteration
- Feller process
- Fredholm map
Theorems and results 62
- Nonemptiness of the spectrum
- Cotlar–Stein lemma
- Riesz–Thorin theorem
- Shift operator as an exponential of differentiation
- Szegő's theorem
- Young's inequality for integral operators
- Bounded inverse theorem
- Calderón–Vaillancourt theorem
- Compactness of the spectrum
- Landweber iteration
- Löwner's theorem
- Noether theorems for singular integral equations
- Peetre theorem on local operators
- Picard's criterion for first-kind equations
- Spectral mapping theorem for polynomials
- Szegő limit theorems
- T(1) theorem
- Toeplitz index theorem
- Abian–Brown theorem
- Closed graph theorem
- Coifman–Rochberg–Weiss commutator theorem
- Convergence of approximations for second-kind equations
- Knaster–Tarski theorem
- Ky Fan minimax inequality
- Lions–Stampacchia theorem
- Operator Cauchy equation and one-parameter semigroups
- Existence of an eigenvalue for a nonzero symmetric kernel
- Gelfand's spectral radius formula
- Spectral theorem for normal matrices
- von Neumann mean ergodic theorem
- Davis's converse of the Knaster–Tarski theorem
- Palais–Smale theory of critical points
- Eigenvalues of a self-adjoint operator are real
- An iterate of the Picard operator is a contraction
- Monotone operator method for Hammerstein equations
- Eigenspaces of a self-adjoint operator are orthogonal
- Fredholm property of elliptic operators
- Wiener–Hopf factorization
- Matrix of the adjoint is the conjugate transpose
- Fredholm's theorem
- Bourbaki–Witt theorem
- Bounded linear maps are continuous
- Ill-posedness of first-kind equations
- Stein interpolation theorem
- Bilinear series of a symmetric kernel
- Hilbert–Schmidt expansion theorem
- Resolvent as Laplace transform of a semigroup
- Courant–Fischer characterization of characteristic values
- Matrix Wiener–Hopf factorization
- Contraction mapping principle for integral equations
- Schmidt's singular value decomposition of a kernel
- Mercer's theorem
- Inverse function theorem in Banach spaces
- Implicit function theorem in Banach spaces
- Nash–Moser theorem
- Lidskii's trace formula for integral operators
- Linear maps on finite-dimensional spaces are continuous
- Rellich–Kondrachov theorem
- Schauder fixed-point theorem for integral equations
- Compactness of integral operators
- Lax–Milgram theorem
- Jentzsch's theorem on positive kernels
Problems 3
Definitions 150
- Operator system
- Nonexpansive map
- Coercive bilinear form
- Crossed product
- Densely defined operator
- Dissipative operator
- Fixed-point property of posets
- Inverse problem
- Krein space
- Least fixed point
- Leray–Schauder degree
- Monotone operator
- Nilpotent operator
- Perturbation theory of linear operators
- Pincherle derivative
- Positive element
- Positive operator
- Singular integral operator
- Square root of a positive operator
- Subnormal operator
- Symmetric operator
- Unitary operator
- Analytic semigroup
- Approximate point spectrum
- Beurling–Ahlfors transform
- Calderón–Zygmund operator
- Characteristic value of a kernel
- Contraction semigroup
- Determinant line bundle
- Eigenfunction of an integral operator
- Elliptic partial differential equation
- Essential spectrum
- Fredholm determinant
- Generator of a semigroup
- Graph of an operator
- Hilbert–Schmidt integral operator
- Hilbert–Schmidt operator
- Hyponormal operator
- Index of a singular integral operator
- Index theory
- Matrix function
- Nemytskii operator
- Point spectrum
- Positive linear operator
- Positive solutions of integral equations
- Proximal operator
- Pseudo-differential operator
- Resolvent set
- Riesz potential
- Schatten class
- Spectrum of an element
- Trace of an integral operator
- Abstract Volterra equation
- Adjoint of a linear map
- Analytic index
- Anderson localization
- C0-semigroup
- Closable operator
- Closure of an operator
- Collectively compact operator approximation
- Commutant
- Continuous and residual spectrum
- Discrete spectrum
- Elliptic operator
- Finite-rank operator
- Infinitesimal generator of a Markov process
- Invariant subspace
- Loewner order
- Operator space
- Projection method for integral equations
- Projection-valued measure
- Rayleigh quotient for symmetric kernels
- Smale's degree for proper Fredholm maps
- Spectral theory
- Square-integrable kernel
- Strictly singular operator
- Tikhonov regularization
- Unbounded operator
- Weak-type estimate
- Wiener–Hopf equation
- Compact operator
- Critical point theory
- Differential equations in Banach spaces
- Frame operator
- Fredholm index
- Friedrichs extension
- Holomorphic functional calculus
- Koopman operator
- Operator theory
- Palais–Smale compactness condition
- Pseudospectrum
- Reflexive operator algebra
- Regularization of ill-posed problems
- Resolvent
- Restriction of an endomorphism to an invariant subspace
- Spectrum of an operator
- Symbol class
- Bifurcation from a characteristic value
- Differential operator between vector bundles
- Essentially self-adjoint operator
- Fredholm operator
- Ideal of compact operators
- Nest algebra
- Normal operator
- Nuclear operator
- Pick matrix
- Riesz potential operator
- Self-adjoint operator
- Strong operator topology
- Weak operator topology
- Adjoint of an unbounded operator
- Deficiency indices
- Ill-posedness of real Laplace inversion
- Invertible operator
- Numerical range
- Principal symbol of an operator on a manifold
- Ultraweak topology
- Closed operator
- Elliptic operator on a compact manifold
- Leray–Schauder method
- Partial isometry
- Picard operator
- Trace class operator
- Eigenvalue perturbation
- Principal symbol
- Dual operator
- Bessel potential
- Dirac operator
- Fractional Laplacian
- Borel functional calculus
- Shift operator
- Toeplitz operator
- Composition operator
- Contraction mapping
- Riesz projector
- Mild solution
- Lyapunov–Schmidt reduction
- Continuous linear map
- Hilbert space adjoint
- Fixed-point iteration
- Integral operator
- Operator monotone function
- Singular values of a compact operator
- Operator norm
- Wiener–Hopf method
- Hankel operator
- Exponential of a bounded operator
- Feller process
- Fredholm map
- Continuous functional calculus