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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 unitary dilation ''U'' to a Hilbert space ''K'', containing ''H'', with :T^n = P_H U^n \vert_H,\quad n\ge 0. Moreover, such a dilation is unique (up to unitary equivalence) when one assumes ''K'' is minimal, in the sense that the linear span of ∪''n''''UnH'' is dense in ''K''. When this minimality condition holds, ''U'' is called the minimal unitary dilation of ''T''. Proof For a contraction ''T'' (i.e., (\, T\, \le1), its defect operator ''DT'' is defined to be the (unique) positive square root ''DT'' = (''I - T*T'')½. In the special case that ''S'' is an isometry, ''DS*'' is a projector and ''DS=0'', hence the following is an Sz. Nagy unitary dilation of ''S'' with the required polynomial functional calculus property: :U = \begin S & D_ \\ D_S & -S^* \end. Returning to the general case of a contraction ''T'', every contraction ''T'' on a Hilb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Béla Szőkefalvi-Nagy
Béla Szőkefalvi-Nagy (29 July 1913, Kolozsvár – 21 December 1998, Szeged) was a Hungarian mathematician. His father, Gyula Szőkefalvi-Nagy was also a famed mathematician. Szőkefalvi-Nagy collaborated with Alfréd Haar and Frigyes Riesz, founders of the Szegedian school of mathematics. He contributed to the theory of Fourier series and approximation theory. His most important achievements were made in functional analysis, especially, in the theory of Hilbert space operators. He was editor-in-chief of the ''Zentralblatt für Mathematik'', the ''Acta Scientiarum Mathematicarum'', and the ''Analysis Mathematica''. He was awarded the Kossuth Prize in 1953, along with his co-author F. Riesz, for his book ''Leçons d'analyse fonctionnelle.'' He was awarded the Lomonosov Medal in 1979. The Béla Szőkefalvi-Nagy Medal honoring his memory is awarded yearly by Bolyai Institute. His books * Béla Szőkefalvi-Nagy: ''Spektraldarstellung linearer Transformationen des Hilbertschen ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilation (operator Theory)
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] |
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 Szőkefalvi-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 positive semidefinite one given by the spectral theorem. The defect spaces \mathcal_T and \mathcal_ are th ... [...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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ... 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 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] |
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 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 A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a sp ..., an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whe ... [...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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Articles Containing Proofs
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article may also refer to: Government and law * Article (European Union), articles of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of Impeachment, a formal document and charge used for impeachment in the United States * Articles of incorporation, for corporations, U.S. equivalent of articles of association * Articles of organization, for limited liability organizations, a U.S. equivalent of articles of association Other uses * Article, an HTML element, delimited by the tags and * Article of clothing, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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