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Hyperreflexive
In functional analysis, a reflexive operator algebra ''A'' is an operator algebra that has enough invariant subspaces to characterize it. Formally, ''A'' is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in ''A''. This should not be confused with a reflexive space. Examples Nest algebras are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern. In fact if we fix any pattern of entries in an ''n'' by ''n'' matrix containing the diagonal, then the set of all ''n'' by ''n'' matrices whose nonzero entries lie in this pattern forms a reflexive algebra. An example of an algebra which is ''not'' reflexive is the set of 2 × 2 matrices :\left\. This algebra is smaller than the Nest algebra :\left\ but has the same invariant subspaces, so it is not reflexive. If ''T'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nest Algebra
In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a 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 natural ... context. They were introduced by and have many interesting properties. They are non-selfadjoint algebras, are closure (mathematics), closed in the weak operator topology and are reflexive operator algebra, reflexive. Nest algebras are among the simplest examples of commutative subspace lattice algebras. Indeed, they are formally defined as the algebra of bounded operators leaving Invariant (mathematics), invariant each Linear subspace, subspace contained in a subspace nest, that is, a set of subspaces which is total order, totally ordered by subset, inclusion and is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nest Algebra
In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a 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 natural ... context. They were introduced by and have many interesting properties. They are non-selfadjoint algebras, are closure (mathematics), closed in the weak operator topology and are reflexive operator algebra, reflexive. Nest algebras are among the simplest examples of commutative subspace lattice algebras. Indeed, they are formally defined as the algebra of bounded operators leaving Invariant (mathematics), invariant each Linear subspace, subspace contained in a subspace nest, that is, a set of subspaces which is total order, totally ordered by subset, inclusion and is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, 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 function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
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. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...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] |
Reflexive Subspace Lattice
Reflexive may refer to: In fiction: *Metafiction In grammar: *Reflexive pronoun, a pronoun with a reflexive relationship with its self-identical antecedent *Reflexive verb, where a semantic agent and patient are the same In mathematics and computer science: *Reflexive relation, a relation where elements of a set are self-related *Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited *Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it *Reflexive space In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an iso ..., a subset of Banach spaces *Reflexive bilinear form, a bilinear form for which the order of a pair of vectors does not affect whether it evaluates to zero. In biology *Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subspace Lattice
Subspace may refer to: In mathematics * A space inheriting all characteristics of a parent space * A subset of a topological space endowed with the subspace topology * Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication * Flat (geometry), a Euclidean subspace * Affine subspace, a geometric structure that generalizes the affine properties of a flat * Projective subspace, a geometric structure that generalizes a linear subspace of a vector space * Multilinear subspace in multilinear algebra, a subset of a tensor space that is closed under addition and scalar multiplication In science fiction * a concept similar to that of hyperspace In science fiction, hyperspace (also known as nulspace, subspace, overspace, jumpspace and similar terms) is a concept relating to higher dimensions as well as parallel universes and a faster-than-light (FTL) method of interstellar travel. ... * Subspace (''Star Trek''), a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Subspace
In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General description Consider a linear mapping T :T: W \to W. An invariant subspace W of T has the property that all vectors \mathbf \in W are transformed by T into vectors also contained in W. This can be stated as :\mathbf \in W \implies T(\mathbf) \in W. Trivial examples of invariant subspaces * \mathbb^n: Since T maps every vector in \mathbb^n into \mathbb^n. * \: Since a linear map has to map 0 \mapsto 0. 1-dimensional invariant subspace ''U'' A basis of a 1-dimensional space is simply a non-zero vector \mathbf. Consequently, any vector \mathbf \in U can be represented as \lambda \mathbf where \lambda is a scalar. If we represent T by a matrix A then, for U to be an invariant subspace it must satisfy : \forall \mathbf \in U \; \exists \alpha \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type I Von Neumann Algebra
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. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebra, ... [...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] |
Operator Algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are phrased in algebraic terms, while the techniques used are highly analytic.''Theory of Operator Algebras I'' By Masamichi Takesaki, Springer 2012, p vi Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory. Overview Operator algebras can be used to study arbitrary sets of operators with little algebraic relation ''simultaneously''. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings. An operator alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Normal Form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. Let ''V'' be a vector space over a field ''K''. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in ''K'', or equivalently if the characteristic polynomial of the operator splits into linear factors over ''K''. This condition is always satisfied if ''K'' is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
