Discrete Spectrum (mathematics)
In mathematics, specifically in spectral theory, a discrete spectrum of a Unbounded_operator#Closed_linear_operators, 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 (functional analysis), spectrum \sigma(A) of a Unbounded_operator#Closed_linear_operators, closed linear operator A:\,\mathfrak\to\mathfrak in the Banach space \mathfrak with Unbounded_operator#Definitions_and_basic_properties, 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 counterclockwi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 operator (mathematics), operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of System of linear equations, 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 axis theorem, principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Point Spectrum
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 spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 \operatornameT = Y/\operatornameT, and with closed range \operatornameT. The last condition is actually redundant. The ''index'' of a Fredholm operator is the integer : \operatornameT := \dim \ker T - \operatorname\operatornameT or in other words, : \operatornameT := \dim \ker T - \operatorname\operatornameT. 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 \to Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compact operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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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 * 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 \sigma_(T), is the set of all real numbers \lambda \in \R 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.) The definition of essential spectrum \sigma_(T) will remain unchanged if we allow it to consist of all those complex numbers \lambda \in \C (instead of just real numbers) such that the above condition holds. This is due to the fact that the spectrum of self-adjoint cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 \lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Decomposition Of Spectrum (functional Analysis)
The spectrum of a linear operator T that operates on a Banach space X is a fundamental concept of functional analysis. The spectrum 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 operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nilpotent Operator
In operator theory, a bounded operator ''T'' on a Banach space is said to be nilpotent if ''Tn'' = 0 for some positive integer ''n''. It is said to be quasinilpotent or topologically nilpotent if its spectrum ''σ''(''T'') = . Examples In the finite-dimensional case, i.e. when ''T'' is a square matrix (Nilpotent matrix) with complex entries, ''σ''(''T'') = if and only if ''T'' is similar to a matrix whose only nonzero entries are on the superdiagonal (this fact is used to prove the existence of Jordan canonical form). In turn this is equivalent to ''Tn'' = 0 for some ''n''. Therefore, for matrices, quasinilpotency coincides with nilpotency. This is not true when ''H'' is infinite-dimensional. Consider the Volterra operator, defined as follows: consider the unit square ''X'' = ,1× ,1⊂ R2, with the Lebesgue measure ''m''. On ''X'', define the kernel function ''K'' by :K(x,y) = \left\{ \begin{matrix} 1, & \mbox{if} \; x \geq y\\ 0, & \mbox{otherwise}. \en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Unbounded Operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as "linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Generalized Eigenvector
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space and let A be the matrix representation of a linear map from V to V with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue \lambda_i is greater than its geometric multiplicity (the nullity of the matrix (A-\lambda_i I), or the dimension of its nullspace). In this case, \lambda_i is called a defective eigenvalue and A is called a defective matrix. A generalized eigenvector x_i corresponding to \lambda_i, together with the matrix (A-\lambda_i I) generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |