In operator theory, the commutant lifting theorem, due to Sz.-Nagy and Foias, is a powerful theorem used to prove several interpolation results.
The commutant lifting theorem states that if is a contraction on a Hilbert space , is its minimal unitary dilation acting on some Hilbert space (which can be shown to exist by Sz.-Nagy's dilation theorem), and is an operator on commuting with , then there is an operator on commuting with such that
and
Here, is the projection from onto . 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.
The commutant lifting theorem can be used to prove the left Nevanlinna-Pick interpolation theorem, the Sarason interpolation theorem, and the two-sided Nudelman theorem, among others.
In mathematics, specifically in functional analysis, a C∗-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product is a linear map A that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
In mathematical physics, the Wightman axioms, named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scattering theory affirmed their significance.
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann.
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.
In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.
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.
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.
In mathematics, affiliated operators were introduced by Murray and von Neumann in the theory of von Neumann algebras as a technique for using unbounded operators to study modules generated by a single vector. Later Atiyah and Singer showed that index theorems for elliptic operators on closed manifolds with infinite fundamental group could naturally be phrased in terms of unbounded operators affiliated with the von Neumann algebra of the group. Algebraic properties of affiliated operators have proved important in L2 cohomology, an area between analysis and geometry that evolved from the study of such index theorems.
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.
The Sz.-Nagy dilation theorem states that every contraction on a Hilbert space has a unitary dilation to a Hilbert space , containing , with
In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.
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.
In complex analysis, given initial data consisting of points in the complex unit disc and target data consisting of points in , the Nevanlinna–Pick interpolation problem is to find a holomorphic function that interpolates the data, that is for all ,
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace.
In mathematics, Hilbert spaces allow the methods of linear algebra and calculus to be generalized 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 induces a distance function for which the space is a complete metric space.
Ciprian Ilie Foiaș was a Romanian-American mathematician. He was awarded the Norbert Wiener Prize in Applied Mathematics in 1995, for his contributions in operator theory.
Donald Erik Sarason was an American mathematician who made fundamental advances in the areas of Hardy space theory and VMO. He was one of the most popular doctoral advisors in the Mathematics Department at UC Berkeley. He supervised 39 Ph.D. theses at UC Berkeley.
This is a glossary for the terminology in a mathematical field of functional analysis.
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