|
Trace Class
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, quantum 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). Definition Let H be a separable Hilbert space, \left\_^ an orthonormal basis and A : H \to H a positive bounded linear operator on H. The trace of A is denoted by \operatorname (A) and defined as :\operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Operator
In mathematical analysis, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions. Motivation On a bounded, smooth domain \Omega \subset \mathbb R^n, consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions: :\begin -\Delta u &= f &\quad&\text \Omega,\\ u &= g &&\text \partial \Omega \end with given functions f and g with regularity discussed in the application section below. The weak solution u \in H^1(\Omega) of this equation must satisfy :\int_\Omega \nabla u \cdot \nabla \varphi \,\mathrm dx = \int_\Omega f \varphi \,\mathrm dx for all \varphi \in H^1_0(\Omega). The H^1(\Omega)-r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). There are several notations, such as \mathbf^\mathrm or \mathbf^*, \mathbf', or (often in physics) \mathbf^. For real matrices, the conjugate transpose is just the transpose, \mathbf^\mathrm = \mathbf^\operatorname. Definition The conjugate transpose of an m \times n matrix \mathbf is formally defined by where the subscript ij denotes the (i,j)-th entry (matrix element), for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\mathbf^\mathrm = \left(\overline\right)^\operatorname = \overline where \mathbf^\operatorname denotes the transpose and \overline denotes the matrix with complex conjugated entries. Other na ... [...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 * either has ''no'' set-theoretic inverse; * or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, I is the identity operator. By the closed graph theorem, \lambda is in the spectrum if and only if the bounded operator T - \lambda I: V\to V is non-bijective on V. 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Linear Operator
In mathematical analysis, an integral linear operator is a linear operator ''T'' given by integration; i.e., :(Tf)(x) = \int f(y) K(x, y) \, dy where K(x, y) is called an integration kernel. More generally, an integral bilinear form is a bilinear functional that belongs to the continuous dual space of X \widehat_ Y, the injective tensor product of the locally convex topological vector spaces (TVSs) ''X'' and ''Y''. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form. These maps play an important role in the theory of nuclear spaces and nuclear maps. Definition - Integral forms as the dual of the injective tensor product Let ''X'' and ''Y'' be locally convex TVSs, let X \otimes_ Y denote the projective tensor product, X \widehat_ Y denote its completion, let X \otimes_ Y denote the injective tensor product, and X \widehat_ Y denote its completion. Suppose that \operatorname : X \otimes_ Y \to X \wid ... [...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] |
Singular Value
In mathematics, in particular functional analysis, the singular values of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T^*T (where T^* denotes the adjoint of T). The singular values are non-negative real numbers, usually listed in decreasing order (''σ''1(''T''), ''σ''2(''T''), …). The largest singular value ''σ''1(''T'') is equal to the operator norm of ''T'' (see Min-max theorem). If ''T'' acts on Euclidean space \Reals ^n, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T (the figure provides an example in \Reals^2). The singular values are the absolute values of the eigenvalues of a normal matrix ''A'', because the spectral theorem can be applied to obtain unitary diagonalization of A as A = U\Lambd ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal (mathematics)
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity'' to linear algebra of bilinear forms. Two elements and of a vector space with bilinear form B are orthogonal when B(\mathbf,\mathbf)= 0. Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. Definitions * In geometry, two Euclidean vectors are orthogonal if they are perpendicular, ''i.e.'' they form a right angle. * Two vectors and in an inner product space V are ''orthogonal'' if their inner product \langle \mathbf, \mathbf \rangle is zero. This relationship is denoted \mathbf \perp \mathbf. * A set of vectors in an inner product space is called pairwise orthogon ... [...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] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for all x, y ∊ ''V''. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
