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Calkin Correspondence
In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its singular value sequence. It originated from John von Neumann's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to (more resolvable) problems on sequence spaces. Definitions A ''two-sided ideal'' ''J'' of the bounded linear operators ''B''(''H'') on a separable Hilbert space ''H'' is a linear subspace such that ''AB'' and ''BA'' belong to ''J'' for all operators ''A'' from ''J'' and ''B'' from ''B''(''H''). A sequence space ''j'' within ''l''∞ can be em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Williams Calkin
John Williams Calkin (11 October 1909, New Rochelle, New York – 5 August 1964, Westhampton, New York) was an American mathematician, specializing in functional analysis. The Calkin algebra is named after him. Biography Calkin received his bachelor's degree from Columbia University in 1933 and his master's degree in 1934 and Ph.D. in 1937 from Harvard University. His doctoral dissertation '' Applications of the Theory of Hilbert Space to Partial Differential Equations; the Self-Adjoint Transformations in Hilbert Space Associated with a Formal Partial Differential Operator of the Second Order and Elliptic Type '') was supervised by Marshall H. Stone. In the dissertation, Calkin acknowledges useful discussions with John von Neumann. At the Institute for Advanced Study, Calkin was a research assistant for the academic year 1937–1938 (working with Oswald Veblen and von Neumann) and in the first eight months of 1942. From 1938 to 1942 he was an assistant professor at the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bra–ket Notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathematically it denotes a vector, \boldsymbol v, in an abstract (complex) vector space V, and physically it represents a state of some quantum system. A bra is of the form \langle f, . Mathematically it denotes a linear form f:V \to \Complex, i.e. a linear map that maps each vector in V to a number in the complex plane \Complex. Letting the linear functional \langle f, act on a vector , v\rangle is written as \langle f , v\rangle \in \Complex. Assume that on V there exists an inner product (\cdot,\cdot) with antilinear first argument, which makes V an inner product space. Then with this inner product each vector \boldsymbol \phi \equiv , \phi\rangle can be identified with a corresponding linear form, by placing the vector in the anti-line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Algebras
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] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schatten Class Operator
In mathematics, specifically functional analysis, a ''p''th Schatten-class operator is a bounded linear operator on a Hilbert space with finite ''p''th Schatten norm. The space of ''p''th Schatten-class operators is a Banach space with respect to the Schatten norm. Via polar decomposition, one can prove that the space of ''p''th Schatten class operators is an ideal in ''B(H)''. Furthermore, the Schatten norm satisfies a type of Hölder inequality: : \, S T\, _ \leq \, S\, _ \, T\, _ \ \mbox \ S \in S_p , \ T\in S_q \mbox 1/p+1/q=1. If we denote by S_\infty the Banach space of compact operators on ''H'' with respect to the operator norm, the above Hölder-type inequality even holds for p \in ,\infty. From this it follows that \phi : S_p \rightarrow S_q ', T \mapsto \mathrm{tr}(T\cdot ) is a well-defined contraction. (Here the prime denotes (topological) dual.) Observe that the ''2''nd Schatten class is in fact the Hilbert space of Hilbert–Schmidt operators ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-rank Operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators on a Hilbert space A canonical form Finite-rank operators are matrices (of finite size) transplanted to the infinite dimensional setting. As such, these operators may be described via linear algebra techniques. From linear algebra, we know that a rectangular matrix, with complex entries, ''M'' ∈ C''n'' × ''m'' has rank 1 if and only if ''M'' is of the form :M = \alpha \cdot u v^*, \quad \mbox \quad \, u \, = \, v\, = 1 \quad \mbox \quad \alpha \geq 0 . Exactly the same argument shows that an operator ''T'' on a Hilbert space ''H'' is of rank 1 if and only if :T h = \alpha \langle h, v\rangle u \quad \mbox \quad h \in H , where the conditions on ''α'', ''u'', and ''v'' are the same as in the finite dimensional case. Therefore, by induction, an operator ''T'' of finite rank ''n'' takes the ... [...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 quest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Space
In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429. are generalisations of the more familiar Lp space, L^ spaces. The Lorentz spaces are denoted by L^. Like the L^ spaces, they are characterized by a norm (mathematics), norm (technically a quasinorm) that encodes information about the "size" of a function, just as the L^ norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the L^ norms, by exponentially rescaling the measure in both the range (p) and the domain (q). The Lorentz norms, like the L^ norms, are invariant under arbitrary rearrangements of the values of a function. Definition The Lorentz space on a measure space (X, \mu) is the space of complex-valued measurable functions f on ''X'' suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence Space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'' of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences like ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Trace
In mathematics, a singular trace is a Von Neumann algebra#Weights, states, and traces, trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite-rank operator, finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of Lp space#The p-norm in countably infinite dimensions, square-summable sequences and spaces of Hilbert space#Examples, square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix ring, matrix algebras have no non-trivial singular traces and the Trace (linear algebra), matrix trace is the unique trace up to scaling. American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of Trace class, trace class operators. Therefore, in distincti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
