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Unitary Dilation
In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''H'' is ''T''. More formally, let ''T'' be a bounded operator on some Hilbert space ''H'', and ''H'' be a subspace of a larger Hilbert space '' H' ''. A bounded operator ''V'' on '' H' '' is a dilation of T if :P_H \; V , _H = T where P_H is an orthogonal projection on ''H''. ''V'' is said to be a unitary dilation (respectively, normal, isometric, etc.) if ''V'' is unitary (respectively, normal, isometric, etc.). ''T'' is said to be a compression of ''V''. If an operator ''T'' has a spectral set X, we say that ''V'' is a normal boundary dilation or a normal \partial X dilation if ''V'' is a normal dilation of ''T'' and \sigma(V)\subseteq \partial X. Some texts impose an additional condition. Namely, that a dilation satisfy the following (calculus) property: :P_H \; f(V) , _H = f(T) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Set
In operator theory, a set X\subseteq\mathbb is said to be a spectral set for a (possibly unbounded) linear operator T on a Banach space if the spectrum of T is in X and von-Neumann's inequality holds for T on X - i.e. for all rational functions r(x) with no poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ... on X :\left\Vert r(T) \right\Vert \leq \left\Vert r \right\Vert_ = \sup \left\ This concept is related to the topic of analytic functional calculus of operators. In general, one wants to get more details about the operators constructed from functions with the original operator as the variable. For a detailed discussion between Spectral Sets and von Neumann's inequality, see. Functional analysis {{Mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x^2 and M an n\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutant Lifting Theorem
In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results. Statement The commutant lifting theorem states that if T is a contraction on a Hilbert space H, U is its minimal unitary dilation acting on some Hilbert space K (which can be shown to exist by Sz.-Nagy's dilation theorem), and R is an operator on H commuting with T, then there is an operator S on K commuting with U such that :R T^n = P_H S U^n \vert_H \; \forall n \geq 0, and :\Vert S \Vert = \Vert R \Vert. Here, P_H is the projection from K onto H. In other words, an operator from the commutant of ''T'' can be "lifted" to an operator in the commutant of the unitary dilation of ''T''. Applications The commutant lifting theorem can be used to prove the left Nevanlinna-Pick interpolation theorem, the Sarason interpolation theorem In mathematics complex analysis, the Sarason interpolation theorem, introduced by , is a generalizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functional Calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ''x'' be a normal element of a C*-algebra ''A'' with an identity element e. Let ''C'' be the C*-algebra of the bounded continuous functions on the spectrum σ(''x'') of ''x''. Then there exists a unique mapping π : C → A, where ''π(f)'' is denoted ''f(x)'', such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = ''x'', where id denotes the function ''z'' → ''z'' on σ(''x''). In particular, this theorem implies that bounded normal operators on a Hilbert space have a continuous functional calculus. Its proof is almost immediate from the Gelfand representation: it suffices to assume ''A'' is the C*-algebra of continuous functions on some compact space ''X'' and define : \pi(f) = f \circ x. Uniqueness follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Algebra
In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space ''X''. It is a closed subalgebra of ''C''(''X''), the uniform algebra of bounded continuous functions on ''X'', whose real parts are dense in the algebra of bounded continuous real functions on ''X''. The concept was introduced by . Example Let \mathcal(X) be the set of all rational functions that are continuous on X; in other words functions that have no poles in X. Then :\mathcal = \mathcal(X) + \overline is a *-subalgebra of C(X), and of C\left(\partial X\right). If \mathcal is dense in C\left(\partial X\right), we say \mathcal(X) is a Dirichlet algebra. It can be shown that if an operator T has X as a spectral set, and \mathcal(X) is a Dirichlet algebra, then T has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem The Sz.-Nagy dilation theorem (proved by Béla Szőkefalvi-Nagy) states that every contraction ''T'' on a Hilbert space ''H'' has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
