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Resolvent Formalism
In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of an operator in the analytic structure of the functional. Given an operator , the resolvent may be defined as : R(z;A)= (A-zI)^~. Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series. The resolvent of can be used to directly obtain information about the spectral decomposition of . For example, suppose is an isolated eigenvalue in the spectrum of . That is, suppose there exists a simple closed curve C_\lambda in the complex plane that separates from the rest of the spectrum of . Then the residue : -\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Theorem On One-parameter Unitary Groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families :(U_)_ of unitary operators that are strongly continuous, i.e., :\forall t_0 \in \R, \psi \in \mathcal: \qquad \lim_ U_t(\psi) = U_(\psi), and are homomorphisms, i.e., :\forall s,t \in \R : \qquad U_ = U_t U_s. Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by , and showed that the requirement that (U_t)_ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impressive result, as it allows one to define the derivative of the mapping t \mapsto U_t, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras. Formal statem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolvent Set
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions Let ''X'' be a Banach space and let L\colon D(L)\rightarrow X be a linear operator with domain D(L) \subseteq X. Let id denote the identity operator on ''X''. For any \lambda \in \mathbb, let :L_ = L - \lambda\,\mathrm. A complex number \lambda is said to be a regular value if the following three statements are true: # L_\lambda is injective, that is, the corestriction of L_\lambda to its image has an inverse R(\lambda, L)=(L-\lambda \,\mathrm)^ called the resolvent; # R(\lambda,L) is a bounded linear operator; # R(\lambda,L) is defined on a dense subspace of ''X'', that is, L_\lambda has dense range. The resolvent set of ''L'' is the set of all regular values of ''L'': :\rho(L) = \. The spectrum is the complement of the resolve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accumulation Point
In mathematics, a limit point, accumulation point, or cluster point of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood of x contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence (x_n)_ in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n \in V. This definition of a cluster or accumulation point of a sequence generalizes to Net (mathematics), nets and Filter (set theory), filters. The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the Convergent sequence, sequence converges to (respectively, the Convergent filter, filter conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements is referred to as A subset \mathcal \subseteq \mathcal that is closed under the involution *, i.e. \mathcal = \mathcal^*, is called A special case of particular importance is the case where \mathcal is a complete normed *-algebra, that satisfies the C*-identity (\left\, a^*a \right\, = \left\, a \right\, ^2 \ \forall a \in \mathcal), which is called a C*-algebra. Especially in the older literature on *-algebras and C*-algebras, such elements are often called Because of that the notations \mathcal_h, \mathcal_H or H(\mathcal) for the set of self-adjoint elements are also sometimes used, even in the more recent literature. Examples * Each positive element of a C*-algebra is * For each element a of a *-algebra, the elements a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact closure in Y). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that X,Y are Banach, but the definition can be extended to more general spaces. Any bounded operator ''T'' that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting. When ''Y'' is a Hilbert space, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the norm topology. Whether this was true in general for Banach spaces (the approximation property) was an unsolved que ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning 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, mathematical formulation of quantum mechanics, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbounded 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dunford And Schwartz
''Linear Operators'' is a three-volume textbook on the theory of linear operators, written by Nelson Dunford and Jacob T. Schwartz. The three volumes are (I) ''General Theory''; (II) ''Spectral Theory, Self Adjoint Operators in Hilbert Space''; and (III) ''Spectral Operators''. The first volume was published in 1958, the second in 1963, and the third in 1971. All three volumes were reprinted by Wiley in 1988. Canonically cited as Dunford and Schwartz, the textbook has been referred to as "the definitive work" on linear operators. The work began as a written set of solutions to the problems for Dunford's graduate course in linear operators at Yale. Schwartz, a prodigy, had taken his undergraduate degree at Yale in 1948, age 18. In 1949 he began his graduate studies and enrolled in his course. Dunford recognised Schwartz's intelligence and they began a long collaboration, with Dunford acting as Schwartz's advisor for his dissertation ''Linear Elliptic Differential Operators''. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolvent Set
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions Let ''X'' be a Banach space and let L\colon D(L)\rightarrow X be a linear operator with domain D(L) \subseteq X. Let id denote the identity operator on ''X''. For any \lambda \in \mathbb, let :L_ = L - \lambda\,\mathrm. A complex number \lambda is said to be a regular value if the following three statements are true: # L_\lambda is injective, that is, the corestriction of L_\lambda to its image has an inverse R(\lambda, L)=(L-\lambda \,\mathrm)^ called the resolvent; # R(\lambda,L) is a bounded linear operator; # R(\lambda,L) is defined on a dense subspace of ''X'', that is, L_\lambda has dense range. The resolvent set of ''L'' is the set of all regular values of ''L'': :\rho(L) = \. The spectrum is the complement of the resolve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
