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Extensions Of Symmetric Operators
In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems. This article discusses a few related problems of this type. The unifying theme is that each problem has an operator-theoretic characterization which gives a corresponding parametrization of solutions. More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometry, partial isometries. Symmetric operators Let ''H'' be a Hilbert space. A linear operator ''A'' acting on ''H'' with dense domain Dom(''A'') is symmetric if :\langle ... [...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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. 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 continuous function, continuous, unitary operator, unitary etc. 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 variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closable Operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as " linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the ... [...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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. 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 continuous function, continuous, unitary operator, unitary etc. 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 variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accretive Operator
In mathematics, a dissipative operator is a linear operator ''A'' defined on a linear subspace ''D''(''A'') of Banach space ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'') :\, (\lambda I-A)x\, \geq\lambda\, x\, . A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''. An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator. The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups. Properties A dissipative operator has the following properties: * From the inequality g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrichs Extension
In functional analysis, the Friedrichs extension is a canonical form, canonical self-adjoint operator, self-adjoint extension of a non-negative densely defined self-adjoint operator#Symmetric operators, symmetric operator. It is named after the mathematician Kurt Friedrichs. This extension is particularly useful in situations where an operator may fail to be essentially self-adjoint or whose essential self-adjointness is difficult to show. An operator ''T'' is non-negative if : \langle \xi \mid T \xi \rangle \geq 0 \quad \xi \in \operatorname\ T Examples Example. Multiplication by a non-negative function on an ''L''2 space is a non-negative self-adjoint operator. Example. Let ''U'' be an open set in R''n''. On ''L''2(''U'') we consider differential operators of the form : [T \phi](x) = -\sum_ \partial_ \ \quad x \in U, \phi \in \operatorname_c^\infty(U), where the functions ''a''''i j'' are infinitely differentiable real-valued functions on ''U''. We consider ''T'' actin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Completion
Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry (i,j) represents the rating of movie j by customer i, if customer i has watched movie j and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. Another example is the document-term matrix: The frequencies of words used in a collection of documents can be represented as a matrix, where each entry corresponds to the number of times the associated term appears in the indicated document. Without any restrictions on the number of degrees of freedom in the completed matrix this problem is underdetermined since the hidden entries could be assigned arbitrary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction (operator Theory)
In operator theory, a bounded operator ''T'': ''X'' → ''Y'' between normed vector spaces ''X'' and ''Y'' is said to be a contraction if its operator norm , , ''T'' , , ≤ 1. This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla SzÅ‘kefalvi-Nagy and Ciprian Foias. Contractions on a Hilbert space If ''T'' is a contraction acting on a Hilbert space \mathcal, the following basic objects associated with ''T'' can be defined. The defect operators of ''T'' are the operators ''DT'' = (1 − ''T*T'')½ and ''DT*'' = (1 − ''TT*'')½. The square root is the square root of a matrix, positive semidefinite one given by the spectral theorem. The defect spaces \mathcal_T a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Operator
In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: \hat = - i \hbar \frac where is Planck's reduced constant, the imaginary unit, is the spatial coordinate, and a partial derivative (denoted by \partial/\partial x) is used instead of a total derivative () since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows: \hat\psi = - i \hbar \frac In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by , i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Continuous Function
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus— differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the '' Radon–Nikodym derivative'', or ''density'', of a measure. We have the following chains of inclusions for functions over a compact subset of the real line: : ''absolutely continuous'' ⊆ ''uniformly continuous'' = ''cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Transform
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . Real homography The Cayley transform is an automorphism of the real projective line that permutes the elements of in sequence. For example, it maps the positive real numbers to the interval ˆ’1, 1 Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is : ,\ 1= \left frac ,\ 1\right\thicksim - 1, \ x + 1= ,\ 1begin1 & 1 \\ -1 & 1 \end . Complex homography On the Riemann sphere, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Densely Defined
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they ''a priori'' "make sense". Definition A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace \operatorname(T) of X and takes values in Y, written T : \operatorname(T) \subseteq X \to Y. Sometimes this is abbreviated as T : X \to Y when the context makes it clear that X might not be the set-theoretic domain of T. Examples Consider the space C^0(, 1 \R) of all real-valued, continuous functions defined on the unit interval; let C^1(, 1 \R) denote the subspace consisting of all c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
