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Continuous Functional Calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ''x'' be a normal element of a C*-algebra ''A'' with an identity element e. Let ''C'' be the C*-algebra of the bounded continuous functions on the spectrum σ(''x'') of ''x''. Then there exists a unique mapping π : C → A, where ''π(f)'' is denoted ''f(x)'', such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = ''x'', where id denotes the function ''z'' → ''z'' on σ(''x''). In particular, this theorem implies that bounded normal operators on a Hilbert space have a continuous functional calculus. Its proof is almost immediate from the Gelfand representation: it suffices to assume ''A'' is the C*-algebra of continuous functions on some compact space ''X'' and define : \pi(f) = f \circ x. Uniqueness follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") 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: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x^2 and M an n\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are * unitary operators: ''N*'' = ''N−1'' * Hermitian operators (i.e., self-adjoint operators): ''N*'' = ''N'' * Skew-Hermitian operators: ''N*'' = −''N'' * positive operators: ''N'' = ''MM*'' for some ''M'' (so ''N'' is self-adjoint). A normal matrix is the matrix expression of a normal operator on the Hilbert space C''n''. Properties Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable. Let T be a bounded operator. The following are equivalent. * T is normal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (functional Analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is said to be in the spectrum of a bounded linear operator T if T-\lambda I is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator ''R'' on the Hilbert space ℓ2, :(x_1, x_2, \dots) \mapsto (0, x_1, x_2, \dots). This has no eigenvalues, since if ''Rx''=''λx'' then by expanding this expression we see ... [...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] |
Gelfand Representation
In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: * a way of representing commutative Banach algebras as algebras of continuous functions; * the fact that for commutative C*-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras ''L''1(R) and \ell^1() whose translates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Weierstrass Theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof . His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X are shown to suffice, as is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Functional Calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if ''T'' is an operator, applying the squaring function ''s'' → ''s''2 to ''T'' yields the operator ''T''2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator or the exponential e^. The 'scope' here means the kind of ''function of an operator'' which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus one. More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. Motivation If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Functional Calculus
In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ''f'' of a complex argument ''z'' and an operator ''T'', the aim is to construct an operator, ''f''(''T''), which naturally extends the function ''f'' from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of ''T'' to the bounded operators. This article will discuss the case where ''T'' is a bounded linear operator on some Banach space. In particular, ''T'' can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction. Motivation Need for a general functional calculus In this section ''T'' will be assumed to be a ''n'' × ''n'' matrix with complex entries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
