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Topics referred to by the same term
Hilbert operator may refer to: The epsilon operator in Hilbert's epsilon calculus The Hilbert–Schmidt operators on a Hilbert space Hilbert–Schmidt integral
Hilbert_operator
Topic in mathematics
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A : H → H {\displaystyle A\colon H\to
Hilbert–Schmidt_operator
Type of vector space in math
The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from
Hilbert_space
Surjective bounded operator on a Hilbert space preserving the inner product
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include
Unitary_operator
Functional analysis concept
operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Type o integral transform in mathematics
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain Ω in Rn, any k : Ω × Ω → C such that
Hilbert–Schmidt integral operator
Hilbert–Schmidt_integral_operator
Linear operator equal to its own adjoint
applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum
Self-adjoint_operator
Function acting on the space of physical states in physics
one) in a special complex Hilbert space. Time evolution in this vector space is given by the application of the evolution operator. Any observable, i.e.,
Operator_(physics)
Integral transform and linear operator
transform is a bounded operator on L p ( R ) {\displaystyle L^{p}(\mathbb {R} )} for 1 < p < ∞, and that similar results hold for the Hilbert transform on the
Hilbert_transform
(on a complex Hilbert space) continuous linear operator
functional analysis, a normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H → H {\displaystyle N\colon
Normal_operator
Mathematical conjecture about the Riemann zeta function
the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is
Hilbert–Pólya_conjecture
In mathematics, a linear operator acting on inner product space
positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space
Positive_operator
Conjugate transpose of an operator in infinite dimensions
transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition
Hermitian_adjoint
Tensor product space endowed with a special inner product
product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
German mathematician (1862–1943)
arithmetic of ends Hilbert's paradox of the Grand Hotel Hilbert–Schmidt operator Hilbert–Smith conjecture Hilbert–Burch theorem Hilbert's irreducibility theorem
David_Hilbert
Compact operator for which a finite trace can be defined
"trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such
Trace_class
Type of continuous linear operator
trace-class operator is Hilbert–Schmidt, and every Hilbert–Schmidt operator is compact. The converses are false in general. For example, an operator with singular
Compact_operator
Mathematical study of linear operators
space on which the operator acts. A normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H → H {\displaystyle
Operator_theory
Linear operator
The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers
Jacobi_operator
Result about when a matrix can be diagonalized
perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral
Spectral_theorem
*-algebra of bounded operators on a Hilbert space
*-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type
Von_Neumann_algebra
Topics referred to by the same term
In mathematics, Hilbert–Schmidt may refer to a Hilbert–Schmidt operator; a Hilbert–Schmidt integral operator; the Hilbert–Schmidt theorem. This disambiguation
Hilbert–Schmidt
In functional analysis, the Hilbert–Carleman determinant is an operator determinant for certain integral operators on Banach spaces, whose kernels are
Hilbert–Carleman_determinant
Construction for adding objects to a Hilbert space
using the Hilbert space of square-integrable functions on the real line, eigenstates of the position and momentum operators are not in the Hilbert space,
Rigged_Hilbert_space
operator which is functioning under proper operator behavior. T on a Hilbert space H is an operator on a larger Hilbert space K, whose restriction to H composed
Dilation_(operator_theory)
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible
Kuiper's_theorem
Branch of functional analysis
to algebras of operators on a separable Hilbert space, endowed with the operator norm topology. In the case of operators on a Hilbert space, the Hermitian
Operator_algebra
especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples
Subnormal_operator
Bounded operators with sub-unit norm
insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian
Contraction_(operator_theory)
In functional analysis, a Hilbert space
kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Topologies on operators on a Hilbert space
arrows pointing from strong to weak. If H is a Hilbert space, the linear space of Hilbert space operators B(X) has a (unique) predual B ( H ) ∗ {\displaystyle
Operator_topologies
Measure of the "size" of linear operators
operators Topologies on the set of operators on a Hilbert space Unbounded operator – Linear operator defined on a dense linear subspace (Bhatia 1997, p
Operator_norm
Bijective antilinear map between two complex Hilbert spaces
antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be
Antiunitary_operator
Vector space in functional analysis
sets, and the spectrum of an operator on K ~ {\displaystyle {\tilde {K}}} , are understood with respect to this Hilbert space topology. Let L {\displaystyle
Indefinite inner product space
Indefinite_inner_product_space
to obtain information on the operator norm on an operator, acting from one Hilbert space into another, when the operator can be decomposed into almost
Cotlar–Stein_lemma
analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
their counter-part in finite dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace
Nuclear operators between Banach spaces
Nuclear_operators_between_Banach_spaces
conjecture Hilbert–Schmidt inner product Hilbert–Schmidt norm Hilbert–Schmidt operator Hilbert–Schmidt integral operator Hilbert–Schmidt theorem Hilbert–Serre
List of things named after David Hilbert
List_of_things_named_after_David_Hilbert
Generalized normal operator
especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator T on a complex Hilbert space
Hyponormal_operator
bounded operators on a Hilbert space H.". The appropriate morphisms between operator spaces are completely bounded maps. Equivalently, an operator space
Operator_space
the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert
Hilbert–Schmidt_theorem
Locally convex topology on function spaces
mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced
Strong_operator_topology
Notation for quantum states
linear operators (called observables) on the Hilbert space of quantum states. Dynamics are also described by linear operators on the Hilbert space. For
Bra–ket_notation
Generalization of the concept of a direct sum in mathematics
integral or Hilbert integral is a generalization of the concept of a direct sum. The theory is most developed for direct integrals of Hilbert spaces and
Direct_integral
Bounded linear operator
{\displaystyle V(f)(t)=\int _{0}^{t}f(s)\,ds.} V is a bounded linear operator between Hilbert spaces, with kernel form V f ( x ) = ∫ 0 1 1 y ≤ x f ( y ) d y
Volterra_operator
Operators useful in quantum mechanics
realizing the representation as operators on a functional Hilbert space. In the Hilbert space representation case the operators are constructed as follows
Creation and annihilation operators
Creation_and_annihilation_operators
Mathematical objects that generalise the notion of Hilbert spaces
Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in
Hilbert_C*-module
functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator P K T | K : K → K {\displaystyle P_{K}T\vert
Compression (functional analysis)
Compression_(functional_analysis)
Function in mathematical optimization
the proximal operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle
Proximal_operator
Operator in quantum mechanics
x}}} In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication
Momentum_operator
System of formal deduction in logic
a Hilbert system, sometimes called Hilbert calculus, Hilbert-style system, Hilbert-style proof system, Hilbert-style deductive system or Hilbert–Ackermann
Hilbert_system
Linear operator scaling by a fixed function
every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. These operators are often contrasted
Multiplication_operator
Any entity that can be measured
depending on the operator and input. In quantum mechanics, observables correspond to linear self-adjoint operators on a separable complex Hilbert space representing
Observable
It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is a Hilbert–Schmidt operator. The result is also important in
Sazonov's_theorem
Quantum operator for the sum of energies of a system
formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way: The eigenkets of H {\displaystyle H}
Hamiltonian (quantum mechanics)
Hamiltonian_(quantum_mechanics)
Part of Fredholm theories in integral equations
be a Hilbert space with an orthonormal basis { e n } {\displaystyle \{e_{n}\}} indexed by the non negative integers. The unilateral shift operator S on
Fredholm_operator
Operator in probability theory
In probability theory, for a probability measure P on a Hilbert space H with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , the
Covariance_operator
Mathematical norm
that ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), ‖ ⋅ ‖ 1 {\displaystyle \|\cdot \|_{1}} is the trace
Schatten_norm
Mathematical method in functional analysis
with modular conjugation J. This is the special case of Hilbert algebras. The modular operator is trivial and the corresponding von Neumann algebra is
Tomita–Takesaki_theory
Topic in mathematical analysis
In analysis, a branch of mathematics, Hilbert's inequality states that | ∑ r ≠ s u r u s ¯ r − s | ≤ π ∑ r | u r | 2 . {\displaystyle \left|\sum _{r\neq
Hilbert's_inequality
as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt inner product. Abstractly
Liouville_space
Concept in Hilbert spaces mathematics
inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices
Trace_inequality
the set of operators on a Hilbert space norm topology ultrastrong topology strong operator topology weak operator topology weak-star operator topology ultraweak
List of functional analysis topics
List_of_functional_analysis_topics
Operator on a Hilbert space that shifts basis vectors
when acting on the Hilbert space of square-integrable functions. When acting on a measure space, the eigenfunctions of shift operators are characteristically
Unilateral_shift_operator
Partially unsolved problem in mathematics
of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not isomorphic to a separable Hilbert space)
Invariant_subspace_problem
Mathematical concept
In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space H with singular values the same order as the harmonic sequence
Weak_trace-class_operator
Linear operator in functional analysis
\alpha \geq 0.} Exactly the same argument shows that an operator T {\displaystyle T} on a Hilbert space H {\displaystyle H} is of rank 1 {\displaystyle
Finite-rank_operator
Linear operator defined on a dense linear subspace
A densely defined symmetric operator T on a Hilbert space H is called bounded from below if T + a is a positive operator for some real number a. That
Unbounded_operator
Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach
Schatten_class_operator
Mathematical problems related to differential equations
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential
Riemann–Hilbert_problem
Linear operator on dense subset of its apparent domain
the Hilbert space of square-summable sequences, with orthonormal basis ( e n ) n ≥ 1 {\displaystyle (e_{n})_{n\geq 1}} . Define the diagonal operator A
Densely_defined_operator
weak-* topology, or weak-* operator topology or σ-weak topology, is a topology on B(H), the space of bounded operators on a Hilbert space H. B(H) admits a
Ultraweak_topology
Extension of a formal language by the epsilon operator
In logic, Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers
Epsilon_calculus
Mathematical theorem
pairs of points. Associated to K is a linear operator (more specifically a Hilbert–Schmidt integral operator when the interval is compact) on functions
Mercer's_theorem
Topological complex vector space
linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed
C*-algebra
Generalization of a positive-definite matrix
Reproducing kernel Hilbert space Kernel method Berezanskij, Jurij Makarovič (1968). Expansions in eigenfunctions of selfadjoint operators. Providence, RI:
Positive-definite_kernel
Axiomatization of quantum field theory
the Hilbert space are either linear or anti-linear operators (if moreover they preserve the norm, then they are unitary or antiunitary operators); the
Wightman_axioms
Theorem about the dual of a Hilbert space
then H {\displaystyle H} is called a complex Hilbert space (resp. a real Hilbert space). Every real Hilbert space can be extended to be a dense subset of
Riesz_representation_theorem
Mathematical structures that allow quantum mechanics to be explained
mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces. In brief, values of physical observables
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Type of operator in Fourier analysis
many more complicated examples such as the Hilbert transform. In signal processing, a multiplier operator is called a "filter", and the multiplier is
Multiplier_(Fourier_analysis)
Topics referred to by the same term
norm-convergence of a sequence in a Hilbert space (as opposed to weak convergence). The convergence of operators in the strong operator topology. This disambiguation
Strong_convergence
Operator type
especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator T on a complex Hilbert space
Paranormal_operator
Noncommutative geometric structure
linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces
Singular_trace
Description of a quantum-mechanical system
spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector of an observable
Schrödinger_equation
Theorem in functional analysis
variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of
Min-max_theorem
Hungarian-American mathematician (1916–2006)
mathematical logic, probability theory, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as
Paul_Halmos
Formulation of quantum mechanics on a Hilbert Space
give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann
Dirac–von_Neumann_axioms
Identifies the commutant of a specific von Neumann algebra
{\displaystyle {\mathfrak {A}}} . The Hilbert–Schmidt operators on an infinite-dimensional Hilbert space form a Hilbert algebra with inner product (a, b)
Commutation theorem for traces
Commutation_theorem_for_traces
Mathematical description of quantum state
wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described
Wave_function
Theory in probability theory
}^{-1/2}C_{\nu }^{1/2})(C_{\mu }^{-1/2}C_{\nu }^{1/2})^{\ast }-I} is a Hilbert–Schmidt operator on H ¯ . {\displaystyle {\bar {H}}.} A simple consequence of the
Feldman–Hájek_theorem
Symbol connecting formulas in logic
logical connective (also called a logical operator, sentential connective, or sentential operator) is an operator that combines or modifies one or more logical
Logical_connective
Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence
Von Neumann bicommutant theorem
Von_Neumann_bicommutant_theorem
Theory in quantum mechanics
the lattice of logical propositions and the lattice of projection operators on a Hilbert space (See quantum logic). The HPO formalism is a natural extension
HPO_formalism
Kind of linear transformation
space of bounded linear operators L ( H ) {\displaystyle L(H)} on a Hilbert space H becomes a C*-algebra and especially an operator space. It is possible
Bounded_operator
Operation on self-adjoint operators
functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes
Extensions of symmetric operators
Extensions_of_symmetric_operators
Process in quantum mechanical theories
Observables are represented by operators acting on a Hilbert space of such quantum states. The eigenvalue of an operator acting on one of its eigenstates
Canonical_quantization
Mathematical entity to describe the probability of each possible measurement on a system
complex Hilbert space, while each measurable physical quantity (such as the energy or momentum of a particle) is associated with a mathematical operator called
Quantum_state
Mathematical tool in quantum physics
semi-definite operator, see below. A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the
Density_matrix
Operator in quantum mechanics
the Hilbert space of complex-valued, square-integrable functions on the real line. The position operator is defined as the self-adjoint operator Q : D
Position_operator
Measure used in functional analysis
respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results
Projection-valued_measure
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