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Spectral Theory Of Compact Operators
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 space ''H'', the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do not appear in the finite-dimensional case, i.e. for matrices. The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator. In particular, the spectral properties of compact operators resemble those of square matrices. This article first summarizes the corresponding results from the matrix case before discussing the spectral properties of compact operators. The reader will see that most statements transfer verbatim from the matrix case. The spectral theory of compact operators was first developed by F. Riesz. Spectral theory of matrices The classical result for squa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that X,Y are Banach, but the definition can be extended to more general spaces. Any bounded operator ''T'' that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ''Y'' is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved que ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Compact Set
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (functional Analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is said to be in the spectrum of a bounded linear operator T if T-\lambda I * either has ''no'' set-theoretic inverse; * or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, I is the identity operator. By the closed graph theorem, \lambda is in the spectrum if and only if the bounded operator T - \lambda I: V\to V is non-bijective on V. The study of spectra and related properties is known as ''spectral theory'', which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz's Lemma
In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when the normed space is not an inner product space. Statement If X is a reflexive Banach space then this conclusion is also true when \alpha = 1. Metric reformulation As usual, let d(x, y) := \, x - y\, denote the canonical metric induced by the norm, call the set \ of all vectors that are a distance of 1 from the origin , and denote the distance from a point u to the set Y \subseteq X by d(u, Y) ~:=~ \inf_ d(u, y) ~=~ \inf_ \, u - y\, . The inequality \alpha \leq d(u, Y) holds if and only if \, u - y\, \geq \alpha for all y \in Y, and it formally expresses the notion that the distance between u and Y is at least \alpha. Because every vector subspace (such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor–Bendixson Theorem
In the mathematical field of descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space ''X'' have the perfect set property in a particularly strong form: any closed subset of ''X'' can be written uniquely as the disjoint union of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Functional Calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''(''T''), which naturally extends the function ''f'' from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of ''T'' to the bounded operators. This article will discuss the case where ''T'' is a bounded linear operator on some Banach space. In particular, ''T'' can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction. Motivation Need for a general functional calculus In this section ''T'' will be assumed to be a ''n'' × ''n'' matrix with complex entrie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection Operator
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P\colon V \to V such that P^2 = P. When V has an inner product and is complete, i.e. when V is a Hilbert space, the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theorem
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory Of Normal C*-algebras
In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra \mathcal(H) of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of \mathcal(H). A subalgebra A of \mathcal(H) is called normal if it is commutative and closed under the \ast operation: for all x,y\in A, we have x^\ast\in A and that xy = yx. Resolution of identity Throughout, H is a fixed Hilbert space. A projection-valued measure on a measurable space (X, \Omega), where \Omega is a σ-algebra of subsets of X, is a mapping \pi : \Omega \to \mathcal(H) such that for all \omega \in \Omega, \pi(\omega) is a self-adjoint projection on H (that is, \pi(\omega) is a bounded linear operator \pi(\omega) : H \to H that satisfies \pi(\omega) = \pi(\omega)^* and \pi(\omega) \circ \pi(\omega) = \pi(\omega)) such that \pi(X) = \operatorname_H \quad (where \operatorname_H is the identity operator of H) and for every x, y \in H, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of System of linear equations, systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on Principal axis theorem, principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
