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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] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a partial order. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", " ... [...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 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] |
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] |
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] |
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] |
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] |
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". A closed operator that is used in practice is often densely defined. 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 ... [...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] |
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] |
Discrete Spectrum
In the physical sciences, the term ''spectrum'' was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersion (optics), dispersed through a prism (optics), prism. Soon the term referred to a plot of light intensity (physics), intensity or power (physics), power as a function of frequency or wavelength, also known as a ''spectral density plot''. Later it expanded to apply to other waves, such as sound waves and sea waves that could also be measured as a function of frequency (e.g., noise spectrum, sea wave spectrum). It has also been expanded to more abstract "signals", whose power spectrum can be spectrum analyzer, analyzed and signal processing, processed. The term now applies to any signal that can be measured or decomposed along a continuous variable, such as energy in electron spectroscopy or mass-to-charge ratio in mass spectrometry. Spectrum is also used to refer to a graphical representation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
