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Lumer–Phillips Theorem
In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup. Statement of the theorem Let ''A'' be a linear operator defined on a linear subspace ''D''(''A'') of the Banach space ''X''. Then ''A'' generates a contraction semigroup if and only if # ''D''(''A'') is dense in ''X'', # ''A'' is closed, # ''A'' is dissipative, and # ''A'' − ''λ''0''I'' is surjective for some ''λ''0> 0, where ''I'' denotes the identity operator. An operator satisfying the last two conditions is called maximally dissipative. Variants of the theorem Reflexive spaces Let ''A'' be a linear operator defined on a linear subspace ''D''(''A'') of the reflexive Banach space ''X''. Then ''A'' generates a contraction semigroup if and only if # ''A'' is dissipative, and # ... [...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] |
Surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...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] |
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] |
Sobolev Space
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. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Operator
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where \langle \cdot,\cdot \rangle is the inner product on the vector space. The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H. The definition has been further extended to include unbounded '' densely defined'' operators whose domain is topologically dense in—but not necessarily equal to—H. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Operator
Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), an American slasher film * ''Identity'' (game show), an American game show * ''Identity'' (TV series), a British police procedural drama television series * "Identity" (''Arrow''), a 2013 episode * "Identity" (''Burn Notice''), a 2007 episode * "Identity" (''Charlie Jade''), a 2005 episode * "Identity" (''Legend of the Seeker''), a 2008 episode * "Identity" (''Law & Order: Special Victims Unit'' episode), 2005 * "Identity" (''NCIS: Los Angeles''), a 2009 pilot episode Music Albums * ''Identity'' (3T album), 2004 * ''Identity'' (BoA album), 2010 * ''Identity'' (Far East Movement album), 2016 * ''Identity'' (Robert Pierre album), 2008 * ''Identity'' (Raghav album), 2008 * ''Identity'' (Victon EP), 2017 * ''Identity'' (Zee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dissipative Operator
In mathematics, a dissipative operator is a linear operator ''A'' defined on a linear subspace ''D''(''A'') of Banach space ''X'', taking values in ''X'' such that for all ''λ'' > 0 and all ''x'' ∈ ''D''(''A'') :\, (\lambda I-A)x\, \geq\lambda\, x\, . A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all ''λ'' > 0 the operator ''λI'' − ''A'' is surjective, meaning that the range when applied to the domain ''D'' is the whole of the space ''X''. An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator. The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups. Properties A dissipative operator has the following properties: * From the inequality g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Günter Lumer
Günter Lumer (1929–2005) was a mathematician known for his work in functional analysis. He is the namesake of the Lumer–Phillips theorem on semigroups of operators on Banach spaces, and was the first to study L-semi-inner products. Born in Germany and raised in France and Uruguay, he spent his professional career in the United States and Belgium..- . Lumer was born in Frankfurt, Germany, on May 29, 1929. His family fled the Nazis in 1933, moving to France and then again in 1941 to Uruguay, where he became a citizen. Lumer studied at the Universidad de la República, where he came under the influence of Paul Halmos; his first mathematics paper, published in 1953, was jointly authored by Halmos and Juan Jorge Schäffer. He completed a degree in electrical engineering at Montevideo in 1957, and traveled to Halmos' home institution, the University of Chicago, on a Guggenheim Fellowship. At Chicago, he completed a doctorate in 1959 under the supervision of Irving Kaplansky. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as "linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
