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Topic: Hermitian operators

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	Hermitian and Skew Hermitian
		The tranjugate is the transpose of the conjugate.
		Let M be a hermitian matrix and consider x*M.y, where x is a row vector and y is a column vector.
		A skew hermitian operator is linear, and the set of skew hermitian operators on S forms a vector space.
	www.mathreference.com /la-sim,herm.html (825 words)

	[No title]
		- Vanishing theorems on Hermitian manifolds, (with B. Alexandrov), Diff.
		- Dirac operators on Hermitian spin surfaces (with B. Alexandrov), Ann.
		- Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4, (with P.Gauduchon), Mathematische Zeitschrift 226 (1997), 317-326.
	www.fmi.uni-sofia.bg /ivanovsp/works.html (414 words)

	Hermitian : Information and resources about Hermitian : School Work Guru
		In mathematics, a Hermitian matrix is a square matrix with complex that is equal to its own conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:
		A version of the spectral theorem also applies to Hermitian operators; while the eigenvectors to different eigenvalues are orthogonal, in general it is not true that the Hilbert space H admits an orthonormal basis consisting only of eigenvectors of the operator.
		In the mathematical formulation of quantum mechanics, one considers even more general Hermitian operators: they are only defined on a dense subspace of a Hilbert space and don't have to be continuous.
	www.schoolworkguru.org /encyclopedia/h/he/hermitian.html (560 words)

	Hermitian adjoint - Biocrawler (Site not responding. Last check: )
		If one thinks of operators on a Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex conjugate of a complex number.
		Consider a continuous linear operator A : H → H (this is the same as a bounded operator).
		The set of bounded linear operators on a Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C-star algebra.
	www.biocrawler.com /encyclopedia/Hermitian_conjugate (338 words)

	NationMaster - Encyclopedia: Spectral theorem
		In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f.
		This implies the linear operator A − rI is not invertible and hence maps a non-zero vector e to 0.
		In functional analysis, a compact operator (or completely continuous operator) is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a...
	www.nationmaster.com /encyclopedia/Spectral-theorem (3244 words)

	Hermitian adjoint
		Hermitian operators are very important because of their properties.
		The hermitian adjoint of a complex matrix is the transpose of the matrix of the complex conjugates of the elements of the original matrix:
		If an operator is sesquilinear, the hermitian adjoint of that operator is the operator itself.
	www.xasa.com /wiki/en/wikipedia/h/he/hermitian_adjoint.html (369 words)

	Quantum Mechanics (Stanford Encyclopedia of Philosophy)
		A structure is a set of elements on which certain operations and relations are defined, a mathematical structure is just a structure in which the elements are mathematical objects (numbers, sets, vectors) and the operations mathematical ones, and a model is a mathematical structure used to represent some physically significant structure in the world.
		Different operators can have different eigenvectors, but the eigenvector/operator relation depends only on the operator and vectors in question, and not on the particular basis in which they are expressed; the eigenvector/operator relation is, that is to say, invariant under change of basis.
		Hermitian operators in the Hilbert space associated with a system represent physical quantities, and their eigenvalues represent the possible results of measurements of those quantities.
	plato.stanford.edu /entries/qm (3823 words)

	PlanetMath: spectral theorem
		Here are some important classes of normal operators, distinguished by the nature of their eigenvalues.
		In such a context one must distinguish between the so-called discrete and continuous (no corresponding eigenspace) spectrums, and replace the representing sum for the operator with some kind of an integral.
		Finally, there are versions of the spectral theorem, of importance in theoretical quantum mechanics, that can be applied to continuous 1-parameter groups of commuting, self-adjoint operators.
	planetmath.org /encyclopedia/SpectralTheoremForHermitianMatrices.html (262 words)

	Hermitian Operators
		All observable properties are represented by Hermitian operators (but not all Hermitian operators correspond to an observable property).
		The eigenstates corresponding to different eigenvalues of hermitian operators are orthogonal.
		This is true of all operators, but also to the hermitian operators of special importance to us here.
	www.cobalt.chem.ucalgary.ca /ziegler/educmat/chm386/rudiment/quanmath/hermit.htm (140 words)

	commutator of hermitian operators
		I understand how C is hermitian as well, but i can't figure out how to prove the equation.
		=> ([A,B])^+= BA - AB=[B,A]=-[A,B] => the commutator of hermitian operators is an anti hermitian operator.
		And an antihermitian operator is an hermitian operator times i.
	www.physicsforums.com /showthread.php?goto=lastpost&t=77071 (957 words)

	Regular Extensions of Hermitian Operators
		The concept of regular extensions of an Hermitian (non-denselydefined) operator was introduced by A. Kuzhel in 1980.
		Chapter 2 is devoted to the investigation of regularextensions of Hermitian (symmetric) operators with certainrestrictions.
		In chapter 3 regular extensions of Hermitian operatorswith the use of boundary-value spaces are investigated.
	www.vsppub.com /books/mathe/bk-RegExtHerOpe.html (273 words)

	Hermitian Operators
		An important property of Hermitian operators is that their eigenvalues are real.
		Another important property of Hermitian operators is that their eigenvectors are orthogonal (or can be chosen to be so).
		Thus we have shown that eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal.
	vergil.chemistry.gatech.edu /notes/quantrev/node16.html (172 words)

	Proceedings of the American Mathematical Society
		The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton.
		A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math.
		R.C. Thompson and L.J. Freede, Eigenvalues of partitioned Hermitian matrices, Bulletin Austral.
	www.ams.org /proc/2004-132-10/S0002-9939-04-07072-8/home.html (410 words)

	J.P. Dahl's Book : "Introduction to the Quantum World of Atoms and Molecules." Component of : Early Ideas in the ...
		Emphasis is put on the properties of operators, quantum theory and measurements, and matrix algebra.
		In addition to solving these problems analytically, we also become familiar with wave packets, ladder operators, angular-momentum theory, and the general central-field problem.
		A very fundamental concept in the theory of atoms and molecules is the electron spin.
	www.quantum-chemistry-history.com /Dahl_Dat/DahlBk.htm (1242 words)

	Angular momentum operators
		The first point to note is that expressions (509)-(511) are unambiguous with respect to the order of the terms in multiplicative factors, since the various position and momentum operators appearing in them all commute with one another [see Eqs.
		(509)-(511) are plausible definitions for the quantum mechanical operators which represent the components of angular momentum.
		4.10, that in order for two physical quantities to be (exactly) measured simultaneously, the operators which represent them in quantum mechanics must commute with one another.
	farside.ph.utexas.edu /teaching/qmech/lectures/node70.html (277 words)

	Properties of the D and D operators
		operators are unitary, they are hermitian or antihermitian depending on whether they are involutive or antiinvolutive, respectively.
		Antilinear operators do not give good quantum numbers, and their role is very different, depending on whether they are hermitian or antihermitian, Tables 1 and 2.
		This is in contrast to properties of linear operators, for which a linear combination of eigenstates, corresponding to the same eigenvalue, with arbitrary coefficients, is also an eigenstate with the same eigenvalue.
	info.fuw.edu.pl /~dobaczew/symhfx57w/node7.html (347 words)

	Quantum Mechanics: Matrix Mechanics and Wave Mechanics - Numericana
		Hermitian conjugation generalizes to vectors and operators the complex conjugation of scalars.
		The trace of an operator is the sum of the elements in its main diagonal (this doesn't depend on the base).
		An [analytic] function of an operator, like the logarithm of an operator, is defined in a standard way: In a base where the operator is diagonal, its image is the diagonal operator whose eigenvalues are the images of its eigenvalues.
	home.att.net /~numericana/answer/quantum.htm (3643 words)

	University of Newcastle <ADT> Public View
		The spatial numerical range of an operator on a normed linear space and the algebra numerical range of an element of a unital Banach algebra, as developed by G. Lumer and F. Bonsall, are considered and the theory of such numerical ranges applied to Banach algebra.
		The spatial numerical range of a compact operator, on a Hilbert space or an Lp-space, for p between one and infinity, is shown to contain all the non—zero extreme points of its closure.
		In a Hilbert space the hermitian operators, which attain their numerical radius, are shown to be dense among all the hermitian operators.
	www.newcastle.edu.au /services/library/adt/public/adt-NNCU20030123.144107 (466 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		By the finite-dimensional spectral theorem such operators have an orthonormal basis in which the operator can be represented as a diagonal matrix with entries in the real numbers.
		The spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may not have any eigenvalues.
		Many operators that occur in analysis are bounded below (such as the negative of the Laplacian operator), so the issue of essential adjointness for these operators is less critical.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Self-adjoint_operator (2684 words)

	PlanetMath: self-adjoint operator
		Cross-references: closure, closable, restriction, adjoint, operator, Hilbert space, linear operator, densely defined
		This is version 5 of self-adjoint operator, born on 2003-07-28, modified 2006-06-15.
		(Operator theory :: Special classes of linear operators :: Hermitian and normal operators)
	www.planetmath.org /encyclopedia/HermitianOperator.html (86 words)

	Hermitian adjoint - Gurupedia
		In mathematics, the hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with
		For every linear operator A (which is bounded, see operator's norm), we define the Hermitian adjoint, which we mark by (pronounced "A dagger") by
		Hamiltonian (energy) of physical systems is the most important Hermitian operator, since its govern the developement of the system with time.
	www.gurupedia.com /h/he/hermitian_conjugate.htm (293 words)

	Notes on Diagonalizing Operators
		Just as for Hermitian operators, the left eigenvectors are the adjoint of the right eigenvectors, and the eigenvectors of different eigenvalues are orthogonal.
		If we had a 4 states in a line and imagine a parity operation so that 1 goes to 4, 2 goes to 3, 3 goes to 2, and 4 goes to 1.
		Hermitian and Unitary matrices are never defective, so quantum mechanics is nearly immune to this disease.
	phyastweb.la.asu.edu /phy576-schmidt/diagonalization/index.html (1987 words)

	Hermitian Operators (Site not responding. Last check: )
		Hermitian operators on quantum general relativity loop space...
		A Hermitian operator conjugate to the number operator...
		Remarks on the construction of a Hermitian phase operator...
	www.scienceoxygen.com /phys/151.html (118 words)

	Single-particle bases and matrix structure of the single-particle hermitian operators
		Throughout this section we restrict our analysis to hermitian single-particle operators, and we study their matrix elements in the single-particle space.
		Assumption (9) does not preclude whether or not the time-reversal is a conserved operator; it only defines the property of the single-particle basis in which the dynamic problem is to be solved.
		where A and B are hermitian matrices, Y is arbitrary, and all submatrices are, in general, complex.
	info.fuw.edu.pl /~dobaczew/symhfy57w/node5.html (318 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		This guarantees that various operations, such as orthogonal projection onto a closed subspace, which are easily shown to be defined for finite dimensional Euclidean spaces, are well-defined in general, even for infinite dimensional spaces.
		These operators share many features of the real numbers and are sometimes seen as generalizations of them.
		If a linear operator has a closed graph and is defined on all of a Hilbert space, then, by the closed graph theorem in Banach space theory, it is necessarily bounded.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Hilbert_space (1990 words)

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