In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy
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 measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so continuous.
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system X being in state x as
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. 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 mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
In mathematics, a commutation theorem explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of Hilbert algebras. It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.
In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and Harish-Chandra (1970) introduced a more general c-function called Harish-Chandra's (generalized) C-function. Gindikin and Karpelevich introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.
In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case.
Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term Tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili equation, and related integrable hierarchies. It is a central ingredient in the theory of solitons. Tau functions also appear as matrix model partition functions in the spectral theory of Random Matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers.
