 |
In mathematics, the predual of an object D is an object P whose dual space is D.
For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions.
 | This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it. |
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space. There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly, and may be extended so as to act on related objects. See Operator (physics) for other examples.
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, 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 mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all
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, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.
In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint operator on that space according to the rule
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus, which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential
The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol. Occasionally, the term multiplier operator itself is shortened simply to multiplier. In simple terms, the multiplier reshapes the frequencies involved in any function. This class of operators turns out to be broad: general theory shows that a translation-invariant operator on a group which obeys some regularity conditions can be expressed as a multiplier operator, and conversely. Many familiar operators, such as translations and differentiation, are multiplier operators, although there are many more complicated examples such as the Hilbert transform.
In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
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 mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they a priori "make sense".
In mathematics, Hilbert spaces 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.
In functional analysis, a branch of mathematics, the ultraweak topology, also called the weak-* topology, or weak-* operator topology or σ-weak topology, on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual B*(H) of B(H), the trace class operators on H. In other words it is the weakest topology such that all elements of the predual are continuous (when considered as functions on B(H)).
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Schultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.