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Type of continuous linear operator
mathematics, a compact operator is a linear operator that behaves, in several important respects, like a finite-dimensional operator such as a matrix
Compact_operator
Theory in functional analysis
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert
Spectral theory of compact operators
Spectral_theory_of_compact_operators
Functional analysis concept
compact 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
Technique in mathematics
(A)} such that R ( z ; A ) {\displaystyle R(z;A)} is a compact operator, we say that A has compact resolvent. The spectrum σ ( A ) {\displaystyle \sigma
Resolvent_formalism
Mathematical compact operator
mathematics, a symmetrizable compact operator is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to
Symmetrizable compact operator
Symmetrizable_compact_operator
Type of mathematical space
space. This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906
Compact_space
Topic in mathematics
Hilbert–Schmidt operator T : H → H is a compact operator. A bounded linear operator T : H → H is Hilbert–Schmidt if and only if the same is true of the operator | T
Hilbert–Schmidt_operator
Part of Fredholm theories in integral equations
Fredholm if and only if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator S : Y → X {\displaystyle S:Y\to X} such that
Fredholm_operator
Mathematical theorem
compact operators. The map K ↦ TK is injective. TK is a non-negative symmetric compact operator on L2[a,b]; moreover K(x, x) ≥ 0. To show compactness
Mercer's_theorem
Feature of certain mathematical spaces
definition is that the embedding operator (the identity) i : X → Y {\displaystyle i\colon X\to Y} is a compact operator. Adams, Robert A. (1975). Sobolev
Compact_embedding
Matrix decomposition
{\displaystyle \mathbf {M} } . Compact operators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series
Singular_value_decomposition
Set of eigenvalues of a matrix
functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
holds for compact operators on a Banach space. One restricts to compact operators because every point x in the spectrum of a compact operator T is an eigenvalue;
Jordan_normal_form
Compact operator for which a finite trace can be defined
of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics
Trace_class
Linear operator equal to its own adjoint
Lebesgue measure on [0, ∞). Compact operator on Hilbert space Unbounded operator Hermitian adjoint Normal operator Positive operator Helffer–Sjöstrand formula
Self-adjoint_operator
Linear operator in functional analysis
T {\displaystyle T} is then a compact operator, and one has the canonical form for compact operators. Compact operators are trace class only if the series
Finite-rank_operator
singular operators can be viewed as a generalization of compact operators, as every compact operator is strictly singular. These two classes share some important
Strictly_singular_operator
One of Fredholm's theorems in mathematics
a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. If V
Fredholm_alternative
Topics referred to by the same term
contain them Compact operator, a linear operator that takes bounded subsets to relatively compact subsets, in functional analysis Compact space, a topological
Compact
Topological complex vector space
reference to operators on a Hilbert space. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and
C*-algebra
Topics referred to by the same term
Compactness can refer to: Compact space, in topology Compact operator, in functional analysis Compactness theorem, in first-order logic Compactness measure
Compactness_(disambiguation)
Mathematical study of linear operators
mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may
Operator_theory
operator with continuous symbol f {\textstyle f} and K is a compact operator. Toeplitz operators with continuous symbols commute modulo the compact operators
Toeplitz_algebra
Noncommutative geometric structure
classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative
Singular_trace
integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of
Fredholm_integral_equation
Partially unsolved problem in mathematics
class of polynomially compact operators (operators T {\displaystyle T} such that p ( T ) {\displaystyle p(T)} is a compact operator for a suitably chosen
Invariant_subspace_problem
Mathematical norm
{\displaystyle |T|:={\sqrt {(T^{*}T)}}} , using the operator square root. If T {\displaystyle T} is compact and H 1 , H 2 {\displaystyle H_{1},\,H_{2}} are
Schatten_norm
Kind of linear transformation
bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators. The Laplace operator Δ : H 2 ( R
Bounded_operator
Fredholm operator if and only if T is invertible modulo compact perturbation, i.e. TS = I + C1 and ST = I + C2 for some bounded operator S and compact operators
Atkinson's_theorem
Theorem
Integral operators are not so 'singular'; another way to put it is that for K {\displaystyle K} a continuous kernel, only compact operators are created
Schwartz_kernel_theorem
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
Mathematics lemma in functional analysis
ball in X {\displaystyle X} is compact. In particular, the identity operator on a Banach space X {\displaystyle X} is compact if and only if X {\displaystyle
Riesz's_lemma
Generalization of the exponential function
compact operator for all t ≥ t 0 {\textstyle t\geq t_{0}} ) . The semigroup is called immediately compact if T ( t ) {\textstyle T(t)} is a compact operator
C0-semigroup
Matrix factorisation in mathematics
operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators
Schur_decomposition
Result about when a matrix can be diagonalized
for compact self-adjoint operators is virtually the same as in the finite-dimensional case. Theorem—Suppose A is a compact self-adjoint operator on a
Spectral_theorem
theorem is a result in operator theory due to Hermann Weyl and John von Neumann. It states that, after the addition of a compact operator (Weyl (1909)) or Hilbert–Schmidt
Weyl–von_Neumann_theorem
Theorem in functional analysis
that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of
Min-max_theorem
6-term-sequence. Operator K-theory is a generalization of topological K-theory, defined by means of vector bundles on locally compact Hausdorff spaces
Operator_K-theory
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by
Neumann–Poincaré_operator
Collection of mathematical theories
One can also study the spectral properties of operators on Banach spaces. For example, compact operators on Banach spaces have many spectral properties
Spectral_theory
Generalization of the Perron–Frobenius theorem to Banach spaces
a total cone. Let T : X → X {\displaystyle T:X\to X} be a non-zero compact operator, and assume that it is positive, meaning that T ( K ) ⊂ K {\displaystyle
Krein–Rutman_theorem
Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous
Dunford–Pettis_property
Mathematical result in differential geometry
Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the
Atiyah–Singer_index_theorem
Type of vector space in math
integral equations. Fredholm operators are bounded operators that are invertible modulo compact operators. Thus an operator T {\displaystyle T} is Fredholm
Hilbert_space
Square roots of the eigenvalues of the self-adjoint operator A* A
mathematics, in particular in functional analysis, the singular values of a compact operator T : X → Y {\displaystyle \,T\!:X\rightarrow Y} acting between Hilbert
Singular_value
Linear operators with a common spectrum
not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the
Isospectral
Digital optical disc data storage format
The compact disc (CD) is a digital optical disc data storage format co-developed by Philips and Sony to store and play digital audio recordings. It employs
Compact_disc
_{0})}{\lambda -\lambda _{0}}}} exists for all λ0 ∈ G; and the operator B(λ) is a compact operator for each λ ∈ G. Then either (I − B(λ))−1 does not exist for
Analytic_Fredholm_theorem
Calculus using a logically rigorous notion of infinitesimal numbers
prove that every polynomially compact linear operator on a Hilbert space has an invariant subspace. Given an operator T on Hilbert space H, consider
Nonstandard_analysis
{\displaystyle S_{\infty }} the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds
Schatten_class_operator
Mapping involving integration between function spaces
compact operator acting on a Banach space of functions. Depending on the situation, the kernel is then variously referred to as the Fredholm operator
Integral_transform
Type of differential operator
partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that
Elliptic_operator
Machine
that the operator appears to "ride" the hammer holding the handles like a motorcycle.[citation needed] A small plate compactor A rammer compactor A trench
Compactor
First notice that K is in L2(X, m), therefore T is compact. By the spectral properties of compact operators, any nonzero λ in σ(T) is an eigenvalue. But it
Nilpotent_operator
Theory in mathematics
\rho (a)],(F^{2}-1)\rho (a),(F-F^{*})\rho (a)} for a in A are all B-compact operators. A cycle is said to be degenerate if all three expressions are 0 for
KK-theory
Bounded linear operator
is a Hilbert–Schmidt operator with norm ‖ V ‖ H S 2 = 1 / 2 {\displaystyle \|V\|_{HS}^{2}=1/2} , hence in particular is compact. Its Hermitian adjoint
Volterra_operator
Topics referred to by the same term
refer to: Schauder fixed-point theorem A result about compact operators, see Compact operator § Properties This disambiguation page lists mathematics
Schauder_theorem
Compact embedding theorem concerning Sobolev spaces
completely continuous (compact). Since an embedding is compact if and only if the inclusion (identity) operator is a compact operator, the Rellich–Kondrachov
Rellich–Kondrachov_theorem
Russian-American mathematician (1946–2018)
point theorem, that if a bounded linear operator T on a Banach space commutes with a non-zero compact operator then T has a non-trivial invariant subspace
Victor_Lomonosov
A is the C*-algebra of compact operators on a separable Hilbert space, M(A) is B(H), the C*-algebra of all bounded operators on H. An ideal I in a C*-algebra
Multiplier_algebra
spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators. The commutator subspace of a two-sided
Commutator_subspace
Theorem in linear algebra
type). More generally, it can be extended to the case of non-negative compact operators, which, in many ways, resemble finite-dimensional matrices. These
Perron–Frobenius_theorem
Mathematical theory of integral equations
important results from the general theory is that the kernel is a compact operator when the space of functions are equicontinuous. A related celebrated
Fredholm_theory
Weakly compact operators Finitely strictly singular operators Strictly singular operators Completely continuous operators Pietsch, Albrecht: Operator Ideals
Operator_ideal
Mathematical concept
to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true. Every Hilbert space
Approximation_property
Regularization technique for ill-posed problems
infinite-dimensional context. In the above we can interpret A {\displaystyle A} as a compact operator on Hilbert spaces, and x {\displaystyle x} and b {\displaystyle b}
Ridge_regression
Small construction excavator
Most compact hydraulic excavators have three distinct assemblies: house, undercarriage and workgroup. The house structure contains the operator's compartment
Compact_excavator
Function between topological vector spaces
operator – Kind of linear transformationPages displaying short descriptions of redirect targets Compact operator – Type of continuous linear operator
Continuous_linear_operator
Branch of functional analysis
Those operators in L(X) with similar spectral characteristics are known as Riesz operators. Many classes of Riesz operators (including the compact operators)
Holomorphic functional calculus
Holomorphic_functional_calculus
Differential operator in mathematics
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean
Laplace_operator
Operator generalizing the Laplacian in differential geometry
only operator with this property. As a consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported
Laplace–Beltrami_operator
Magnetic audio tape recording format
The cassette tape, officially named the Compact Cassette, and also known as audio cassette, or simply tape or cassette, is an analog magnetic tape recording
Cassette_tape
Type o integral transform in mathematics
integral operators are both continuous and compact. The concept of a Hilbert–Schmidt integral operator may be extended to any locally compact Hausdorff
Hilbert–Schmidt integral operator
Hilbert–Schmidt_integral_operator
Polish mathematician (1905–1981)
Wiesław Żelazko. List of Polish mathematicians Approximation problem Compact operator Schauder basis Aleksandra Gomułczak, Anna Brożek, Jacek Jadacki, Roman
Stanisław_Mazur
Mathematical tool in quantum physics
representations of A. The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators, and therefore the pure states of K(H) are
Density_matrix
space H, together with a self-adjoint operator F, of square 1 and such that the commutator [F, a] is a compact operator, for all a in A. The paper by Atiyah
Fredholm_module
Information-sharing agreement among US states
Compacts, both of which relate to motor vehicle operator licensing and enforcement. In 1985, draft compacts were developed independently in Colorado and
Interstate Wildlife Violator Compact
Interstate_Wildlife_Violator_Compact
Russian mathematician (born 1940)
University under Viktor Khavin with thesis Invariant subspaces of certain compact operators (title translated from Russian). In 1973 he received his Doctor of
Nikolai_Kapitonovich_Nikolski
Topics referred to by the same term
transcendence Meter Point Administration Number Singular value of a compact operator This disambiguation page lists articles associated with the title S-number
S-number
Linear operator related to topological vector spaces
\Lambda (U)} is precompact in Y. In a Hilbert space, positive compact linear operators, say L : H → H have a simple spectral decomposition discovered
Nuclear_operator
closed unit ball in a normed space is compact in the weak-* topology. adjoint The adjoint of a bounded linear operator T : H 1 → H 2 {\displaystyle T:H_{1}\to
Glossary of functional analysis
Glossary_of_functional_analysis
necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common
Von Neumann bicommutant theorem
Von_Neumann_bicommutant_theorem
stably isomorphic if A ⊗ K ≅ B ⊗ K, where K is the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. C*-algebras are
Hereditary_C*-subalgebra
Aspect of mathematical spectrum theory
essential spectrum is invariant under compact perturbations. That is, if K {\displaystyle K} is a compact self-adjoint operator on X {\displaystyle X} , then
Essential_spectrum
theorem Measure of non-compactness Banach–Mazur theorem Bounded linear operator Continuous linear extension Compact operator Approximation property Invariant
List of functional analysis topics
List_of_functional_analysis_topics
Concept in the solution of linear partial differential equations
is some C ∞ function with compact support. The parametrix is a useful concept in the study of elliptic differential operators and, more generally, of hypoelliptic
Parametrix
ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. Here the addition in B(H)
Calkin_algebra
Typically linear operator defined in terms of differentiation of functions
In mathematics, a differential operator is an operator defined as a function of the differentiation operator (which is an operation that accepts a function
Differential_operator
Sum of elements on the main diagonal
class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm. If K is a trace-class operator, then
Trace_(linear_algebra)
Algebraic structure in linear algebra
differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions
Vector_space
form C*(S) = {Tf + K | Tf is a Toeplitz operator with continuous symbol f ∈ C(T) and K is a compact operator}. In this identification, S = Tz where z
Wold's_decomposition
This visual quality has led to graph C*-algebras being referred to as "operator algebras we can see." Another advantage of graph C*-algebras is that much
Graph_C*-algebra
Computational tool
space K(ℓ2) of compact operators on the Hilbert space ℓ2 has a Schauder basis. For every x, y in ℓ2, let x ⊗ y denote the rank one operator v ∈ ℓ2 → <v,
Schauder_basis
American mathematician
a Fellow of the American Mathematical Society for his contributions to operator theory as well as mentoring and service to the community. He has supervised
Paul_Muhly
Particular kind of algebraic structure
composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on E {\displaystyle E} is a Banach
Banach_algebra
Linear operator in mathematics
mathematics, the composition operator C ϕ {\displaystyle C_{\phi }} with symbol ϕ {\displaystyle \phi } is a linear operator defined by the rule C ϕ ( f
Composition_operator
Binary operator in computer programming
The null coalescing operator is a binary operator that is part of the syntax for a basic conditional expression in several programming languages, such
Null_coalescing_operator
Bottom-up parser that interprets an operator-precedence grammar
an operator-precedence parser is a bottom-up parser that interprets an operator-precedence grammar. For example, most calculators use operator-precedence
Operator-precedence_parser
{\displaystyle A=H(A)} of this operator, which are compact sets. One way of generating such a set is to start with an initial compact set S 0 ⊂ X {\displaystyle
Hutchinson_operator
Compact heavy equipment with differential steering
skid-steer loader (SSL), skid loader, or skidsteer is any of a class of compact heavy equipment with lift arms that can attach to a wide variety of buckets
Skid-steer_loader
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