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Wold Decomposition
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 time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. Details Let ''H'' be a Hilbert space, ''L''(''H'') be the bounded operators on ''H'', and ''V'' ∈ ''L''(''H'') be an isometry. The Wold decomposition states that every isometry ''V'' takes the form :V = (\oplus_ S) \oplus U for some index set ''A'', where ''S'' is the unilateral shift on a Hilbert space ''Hα'', and ''U'' is a unitary operator (possible vacuous). The family consists of isomorphic Hilbert spaces. A proof ... [...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] |
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] |
C*-algebras
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 estab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Subspaces
Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteration * A data type in method overriding that is neither covariant nor contravariant * Class invariant, an invariant used to constrain objects of a class Physics, mathematics, and statistics * Invariant (mathematics), a property of a mathematical object that is not changed by a specific operation or transformation ** Rotational invariance, the property of function whose value does not change when arbitrary rotations are applied to its argument ** Scale invariance, a property of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor ** Topological invariant * Invariant (physics), something does not change under a transformation, such as from one reference frame to another * Invaria ... [...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] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Toeplitz Algebra
In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space ''l''2(N). Taking ''l''2(N) to be the Hardy space ''H''2, the Toeplitz algebra consists of elements of the form :T_f + K\; where ''Tf'' is a Toeplitz operator with continuous symbol and ''K'' is a compact operator. Toeplitz operators with continuous symbols commute modulo the compact operators. So the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators. This extension is called the Toeplitz extension. By Atkinson's theorem, an element of the Toeplitz algebra ''Tf'' + ''K'' is a Fredholm operator if and only if the symbol ''f'' of ''Tf'' is invertible. In that case, the Fredholm index of ''Tf'' + ''K'' is precisely the winding number of ''f'', the equivalence class of ''f'' in the fundamental group of the circle. This is a special case of the Atiyah-Singer index theorem. Wold decomposition c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator On Hilbert Space
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 (representable by finite-dimensional matrices) 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. For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces. More generally, the compactness assumption can be dropped. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Operator
In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. Details Let ''S''1 be the circle, with the standard Lebesgue measure, and ''L''2(''S''1) be the Hilbert space of square-integrable functions. A bounded measurable function ''g'' on ''S''1 defines a multiplication operator ''Mg'' on ''L''2(''S''1). Let ''P'' be the projection from ''L''2(''S''1) onto the Hardy space ''H''2. The ''Toeplitz operator with symbol g'' is defined by :T_g = P M_g \vert_, where " , " means restriction. A bounded operator on ''H''2 is Toeplitz if and only if its matrix representation, in the basis , has constant diagonals. Theorems * Theorem: If g is continuous, then T_g - \lambda is Fredholm if and only if \lambda is not in the set g(S^1). If it is Fredholm, its index is minus the winding number of the curve traced out by g with respect to the origin. For a proof, see . He attributes the theorem to Mark Krein, Harold Widom, and A ... [...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] |
Wandering Set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposite of a conservative system, to which the Poincaré recurrence theorem applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927. Wandering points A common, discrete-time definition of wandering sets starts with a map f:X\to X of a topological space ''X''. A point x\in X is said to be a wandering point if there is a neighbourhood ''U'' of ''x'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
