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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was ther ... [...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 ... [...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] |
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] |
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 spectrum of an operator, spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrix (mathematics), matrices. In broad t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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'' b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Closed Range Theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. History The theorem was proved by Stefan Banach in his 1932 '' Théorie des opérations linéaires''. Statement Let X and Y be Banach spaces, T : D(T) \to Y a closed linear operator whose domain D(T) is dense in X, and T' the transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ... of T. The theorem asserts that the following conditions are equivalent: * R(T), the range of T, is closed in Y. * R(T'), the range of T', is closed in X', the dual of X. * R(T) = N(T')^\perp = \left\. * R(T') = N(T)^\perp = \left\. Where N(T) and N(T') are the null space of T and T', respectively. Corollaries Several ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The ''index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |