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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] |
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] |
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] |
Ciprian Foias
Ciprian Ilie Foiaș (20 July 1933 – 22 March 2020) was a Romanian-American mathematician. He was awarded the Norbert Wiener Prize in Applied Mathematics in 1995, for his contributions in operator theory. Education and career Born in Reșița, Romania, Foias studied mathematics at the University of Bucharest. He completed his dissertation in 1957, but was not allowed to defend his thesis by the Communist government until 1962. He received his doctorate in 1962 under supervision of Miron Nicolescu. Foias defected to France following his lecture at the International Congress of Mathematicians in 1978. He later emigrated to the United States. Foias taught at his alma mater (1966–1979), Paris-Sud 11 University (1979–1983), and Indiana University (1983 until retirement). Beginning in 2000, he was a teacher and researcher at Texas A&M University, where he was a Distinguished Professor. The Foias constant is named after him. Foias is listed as an ISI highly cited researcher. ... [...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 the clo ... [...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] |
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] |
Projection (mathematics)
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: * The projection from a point onto a plane or central projection: If ''C'' is a point, called the center of projection, then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutant
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', or equivalently, such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and normalize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sarason Interpolation Theorem
In mathematics complex analysis, the Sarason interpolation theorem, introduced by , is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation In complex analysis, given ''initial data'' consisting of n points \lambda_1, \ldots, \lambda_n in the complex unit disc \mathbb and ''target data'' consisting of n points z_1, \ldots, z_n in \mathbb, the Nevanlinna–Pick interpolation problem is .... References * Theorems in analysis Interpolation {{mathanalysis-stub ... [...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] |
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