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Hilbert–Schmidt Operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_H, where \ is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm \, \cdot\, _\text is identical to the Frobenius norm. , , ·, , is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if \_ and \_ are such bases, then \sum_i \, Ae_i\, ^2 = \sum_ \left, \langle Ae_i, f_j\rangle \^2 = \sum_ \left, \langle e_i, A^*f_j\rangle \^2 = \sum_j\, A^* f_j\, ^2. If e_i = f_i, then \sum_i \, Ae_i\, ^2 = \sum_i\, A^* e_i\, ^2. As for any bounded operato ... [...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] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are Compact operator, compact operators. In quantum mechanics, Mixed state (physics), mixed states are described by Density matrix, density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Definition Suppose H is a Hilbert s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Inner Product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension - same number of rows and columns, but are not restricted to be square matrices. Definition Given two complex number-valued ''n''×''m'' matrices A and B, written explicitly as : \mathbf = \,, \quad \mathbf = the Frobenius inner product is defined as, : \langle \mathbf, \mathbf \rangle_\mathrm =\sum_\overline B_ \, = \mathrm\left(\overline \mathbf\right) \equiv \mathrm\left(\mathbf^ \mathbf\right) where the overline denotes the complex conjugate, and \dagger denotes Hermitian conjugate. Explicitly this sum is :\begin \langle \mathbf, \mathbf \rangle_\mathrm = & \overline_ B_ + \overline_ B_ + \cdots + \overline_ B_ \\ & + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schatten Norm
In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of ''p''-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Definition Let H_1, H_2 be Hilbert spaces, and T a (linear) bounded operator from H_1 to H_2. For p\in T\, _ = [\mathrm (, T, ^p). If T is compact and H_1,\,H_2 are separable, then : \, T\, _ := \bigg( \sum _ s^p_n(T)\bigg)^ for s_1(T) \ge s_2(T) \ge \cdots s_n(T) \ge \cdots \ge 0 the singular values of T, i.e. the eigenvalues of the Hermitian operator , T, :=\sqrt. Properties In the following we formally extend the range of p to ,\infty/math> with the convention that \, \cdot\, _ is the operator norm. The dual index to p=\infty is then q=1. * The Schatten norms are unitarily invariant: for unitary operators U and V and p\in ,\infty/math>, :: \, U T V\, _p = \, T\, _p. * They satisfy Hölder's inequality: for all p\in ,\infty/math> and q ... [...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] |
Trace Class Operator
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, mixed states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Note that the trace operator studied in partial differential equations is an unrelated concept. Definition Suppose H is a Hilbert space and A : H \to H a bounded linear operator on H which is non-negative (I.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs). Preliminaries and notation Throughout let ''X'',''Y'', and ''Z'' be topological vector spaces (TVSs) and ''L'' : ''X'' → ''Y'' be a linear operator (no assumption of continuity is made unless otherwise stated). * The projective tensor product of two locally convex TVSs ''X'' and ''Y'' is denoted by X \otimes_ Y and the completion of this space will be denoted by X \widehat_ Y. * ''L'' : ''X'' → ''Y'' is a topological homomorphism or homomorphism, if it is linear, continuous, and L : X \to \operatorname L is an open map, where \operatorname L, the image of ''L'', has the subspace topology induced by ''Y''. ** If ''S'' is a subspace of ''X'' then both the quotient map ''X'' → ''X''/''S'' and the canonical injection ''S ... [...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] |
Norm Topology
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the above i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Hilbert Spaces
In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.B. Coecke and E. O. Paquette, Categories for the practising physicist, in: New Structures for Physics, B. Coecke (ed.), Springer Lecture Notes in Physics, 2009arXiv:0905.3010/ref> Definition Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors. Let H_1 and H_2 be two Hilbert spaces with inner products \langle\cdot, \cdot\rangle_1 and \langle\cdot, \cdot\rangle_2, respectively. Cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
