|
Essential Spectrum
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible". The essential spectrum of self-adjoint operators In formal terms, let ''X'' be a Hilbert space and let ''T'' be a self-adjoint operator on ''X''. Definition The essential spectrum of ''T'', usually denoted σess(''T''), is the set of all complex numbers λ such that :T-\lambda I_X is not a Fredholm operator, where I_X denotes the ''identity operator'' on ''X'', so that I_X(x)=x for all ''x'' in ''X''. (An operator is Fredholm if its kernel and cokernel are finite-dimensional.) Properties The essential spectrum is always closed, and it is a subset of the spectrum. Since ''T'' is self-adjoint, the spectrum is contained on the real axis. The essential spectrum is invariant under compact perturbations. That is, if ''K'' ... [...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] |
Orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of An Operator
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] |
Discrete Spectrum (mathematics)
In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank (linear algebra), rank of the corresponding Riesz projector is finite. Definition A point \lambda\in\C in the spectrum \sigma(A) of a closed linear operator A:\,\mathfrak\to\mathfrak in the Banach space \mathfrak with domain \mathfrak(A)\subset\mathfrak is said to belong to ''discrete spectrum'' \sigma_(A) of A if the following two conditions are satisfied: # \lambda is an isolated point in \sigma(A); # The rank (linear algebra), rank of the corresponding Riesz projector P_\lambda=\frac\oint_\Gamma(A-z I_)^\,dz is finite. Here I_ is the identity operator in the Banach space \mathfrak and \Gamma\subset\C is a smooth simple closed counterclockwise-oriented curve bounding an open region \Omega\subset\C such that \lambda is the only point of the spectrum of A in the closure of \Omega; that is, \sigma(A)\cap\overlin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition Of Spectrum (functional Analysis)
The spectrum of a linear operator T that operates on a Banach space X (a fundamental concept of functional analysis) consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectrum has a standard decomposition into three parts: * a point spectrum, consisting of the eigenvalues of T; * a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of T-\lambda a proper dense subset of the space; * a residual spectrum, consisting of all other scalars in the spectrum. This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen. Decomposition into point spectrum, continuous spectrum, and residual spectrum For bounded Banach space operators Let ''X'' be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 \oin ... [...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] |
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] |
Densely Defined Operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they ''a priori'' "make sense". Definition A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace \operatorname(T) of X and takes values in Y, written T : \operatorname(T) \subseteq X \to Y. Sometimes this is abbreviated as T : X \to Y when the context makes it clear that X might not be the set-theoretic domain of T. Examples Consider the space C^0(, 1 \R) of all real-valued, continuous functions defined on the unit interval; let C^1(, 1 \R) denote the subspace consisting of all co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Projector
In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912. Definition Let A be a closed linear operator in the Banach space \mathfrak. Let \Gamma be a simple or composite rectifiable contour, which encloses some region G_\Gamma and lies entirely within the resolvent set \rho(A) (\Gamma\subset\rho(A)) of the operator A. Assuming that the contour \Gamma has a positive orientation with respect to the region G_\Gamma, the Riesz projector corresponding to \Gamma is defined by : P_\Gamma=-\frac\oint_\Gamma(A-z I_)^\,\mathrmz; here I_ is the identity operator in \mathfrak. If \lambda\in\sigma(A) is the only point of the spectrum of A in G_\Gamma, then P_\Gamma is denoted by P_\lambda. Properties The operator P_\Gamma is a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Eigenvalue
In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A-\lambda I has a bounded inverse. The set of normal eigenvalues coincides with the discrete spectrum. Root lineal Let \mathfrak be a Banach space. The root lineal \mathfrak_\lambda(A) of a linear operator A:\,\mathfrak\to\mathfrak with domain \mathfrak(A) corresponding to the eigenvalue \lambda\in\sigma_p(A) is defined as : \mathfrak_\lambda(A)=\bigcup_\\subset\mathfrak, where I_ is the identity operator in \mathfrak. This set is a linear manifold but not necessarily a vector space, since it is not necessarily closed in \mathfrak. If this set is closed (for example, when it is finite-dimensional), it is called the generalized eigenspace of A corresponding to the eigenvalue \lambda. Definition of a normal eigenvalue An eigenvalue \lambda\in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
