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Topic: Adjoint of an operator

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Adjoint

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	Self-adjoint operator - Wikipedia, the free encyclopedia
		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.
		Self-adjoint operators are used in functional analysis and quantum mechanics.
		The operator theoretic adjoint P* of P is a restriction of the distributional extension of the formal adjoint.
	en.wikipedia.org /wiki/Self-adjoint_operator (2722 words)

	Differential operator - Wikipedia, the free encyclopedia
		In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator.
		The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative.
		An operator is a mapping of sections, P: Γ(E) → Γ(F) which maps the stalk of the sheaf of germs of Γ(E) at a point x ∈ M to the fibre of F at x:
	en.wikipedia.org /wiki/Differential_operator (668 words)

	Ji: Toward an exact adjoint (Site not responding. Last check: 2007-10-24)
		An adjoint operator to a forward modeling operator is often a good processing operator for estimating approximate model parameters from data that are assumed to follow the presumed forward modeling operator (Claerbout, 1992).
		The adjoint operator obtained by the first approach has a weighting factor that is the Jacobian, but the one found by the second approach does not have it.
		Even though the discretization of an operator along the data space is more correct than the other approach in simulating the real experiment, we are used to discretizing along the model space because of the ease of numerical implementation.
	sepwww.stanford.edu /public/docs/sep80/jun1/paper_html (2269 words)

	PlanetMath: self-adjoint operator
		is closable and its closure coincides with its adjoint (i.e.
		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.
	planetmath.org /encyclopedia/SelfAdjointOperator.html (86 words)

	[No title] (Site not responding. Last check: 2007-10-24)
		With the advent of the theory of operator spaces, also known as "quantized functional analysis", general, not-necessarily-self-adjoint, operator algebras have achieved an ontological status comparable to that held by the self-adjoint operator algebras.
		Although the non-self-adjoint theory is not yet as well developed, from certain perspectives, the explosion of results that has taken place in the last six years, or so, deserves to be exposed in a fashion that will allow the uninitiated, particularly younger scholars, easy access to the basics of the subject from the current perspective.
		Operator algebra lays bare their basic algebraic properties (how they are manipulated in practice) and metric properties as well (how they are measured).
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9622991.txt (303 words)

	The Support-Operators Method
		The adjoint of an operator varies with the definition of its associated inner products, but is unique for fixed inner products.
		The flux operator is left in the general form of a discrete vector as defined in Step 1.
		Each vector is denoted by an upper-case symbol and the components of that vector are denoted by the corresponding lower-case symbol.
	www.lanl.gov /Augustus/Morel99a/node2.html (2883 words)

	History of Operator Theory
		In the first textbook on operator theory, Théorie des Opérations Linéaires, published in Warsaw 1932, Stefan Banach states that the subject of the book is the study of functions on spaces of infinite dimension, especially those he coyly refers to as spaces of type B, otherwise Banach spaces (definition).
		In 1857 Arthur Cayley introduced the idea of an algebra of matrices, and in 1858 he showed, in modern parlance, that quaternions could be "represented" by matrices.
		Since all these subjects predated operator theory as such by a century or two, it is no surprise that some of the earliest antecedents of operator theory are to be found in them.
	www.mathphysics.com /opthy/OpHistory.html (2635 words)

	FuncAna
		An operator is self-adjoint if and only if its quadratic form is real valued.
		Norm of a self-adjoint operator is equal to the supremum of the absolute value of its quadratic form on the unit sphere.
		The resolvent is an analytic meromorphic function of the spectral parameter.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	What is operator theory? (Site not responding. Last check: 2007-10-24)
		This implies that the dimension of the kernel of an operator is equal to the dimension of the kernel of the adjoint of that operator.
		Operator theory is the study of operators, and the techniques that are used are a combination of algebra and functional analysis.
		More precisely, any normal operator can be approximated, after a unitary transformation, by a diagonal operator plus an operator which is zero except on a finite-dimensional subspace.
	erdos.math.unb.ca /~dan/whatsopth (573 words)

	PlanetMath: adjoint
		is linear, and it is called the adjoint of
		Cross-references: easy to see, operator, implies, linear operator, densely defined, Hilbert space
		This is version 7 of adjoint, born on 2003-07-28, modified 2006-06-15.
	planetmath.org /encyclopedia/Adjoint5.html (61 words)

	Adjoint operators (Site not responding. Last check: 2007-10-24)
		This is because the adjoint operator tolerates imperfections in the data and does not demand that the data provide full information.
		Although inverse operators are more well known than adjoint operators, the inverse is built upon the adjoint so the adjoint is a logical place to start.
		Although the test is an elegant and useful test and is itself a fundamental definition, curiously, that definition does not help construct adjoint operators, so we postpone a formal definition of adjoint until after we have seen many examples.
	sepwww.stanford.edu /sep/prof/bei/conj/paper_html/node1.html (502 words)

	OPERATOR THEORY, SYSTEM THEORY AND SCATTERING THEORY: MULTIDIMENSIONAL GENERALIZATIONS, 2003
		An evident advantage of this approach is a possibility of application of functional analysis methods.
		An effort has been made to include orthogonality of polynomials in several variables with respect to measure supported by an algebraic set into the general framework existing so far (joint work with Dariusz Cicho\'n and Jan Stochel).
		Such a function is an operator, acting on the space of sequences, indexed by the nodes of the tree, and commuting with tree isometries.
	www.cs.bgu.ac.il /~dany/abst4.html (2048 words)

	OPERATOR IDEAL and ARITHMETIC MEANS (Site not responding. Last check: 2007-10-24)
		The bounded operators on a separable infinite dimensional Hilbert space form an algebra, and the two sided ideals of that algebra are called operator ideals.
		The space of commutators of an ideal, which is the natural kernel of the traces on the ideal, has always been of great interest in operator theory, non-commutative geometry, cyclic cohomology and other areas.
		Thus the introduction of the arithmetic mean operation has opened a new chapter in the theory of operator ideals.
	math.uc.edu /news/colloquium/kaftal.htm (159 words)

	[No title]
		This is a kind of decomposition theorem of operators relative to the self-adjoint operator $H$.
		This is defined as follows: Let $T$ be a symemtric operator on ${\cal H}$, $H$ be a self-adjoint operator on ${\cal H}$, and $K(t)$ ($t\in \R$) be a bounded self-adjoint operator on ${\cal H}$ for all $t\in \R$.
		\begin{th}\label{8-1} The operator $H$ is self-adjoint and \begin{equation} H^2=A^2L^2+B^2.\label{eqH2} \end{equation} Moreover \begin{equation} \H\psi\^2=\AL\psi\^2+\B\psi\^2, \quad \psi \in D(H)=D(L)\cap D(B).
	www.ma.utexas.edu /mp_arc/papers/06-36 (6068 words)

	Adjoint Operator for Radiative Transfer (Site not responding. Last check: 2007-10-24)
		The adjoint operator of a constant is itself.
		From equation 5.3 and equation 5.5, the adjoint operator for equation 5.7 is
		Equation 5.11 has the same radiative operator as the regular radiative transfer equation.
	asd-www.larc.nasa.gov /~yhu/paper/thesisall/node39.html (134 words)

	Introduction to Operator Space Theory - Cambridge University Press
		The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory.
		An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory.
		In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra.
	www.cambridge.org /catalogue/catalogue.asp?ISBN=0521811651 (399 words)

	5.1.2 Reverse or adjoint sensitivity
		The reverse nature of the adjoint calculation can be readily seen as follows.
		represents the operator which maps the model state space onto the data space.
		is given by the adjoint operator, driven by the model vs. data misfit:
	mitgcm.org /~adcroft/manual/node165.html (469 words)

	Oxford University Press: Operator Spaces: Edward Effros
		Operator space theory provides a synthesis of Banach space theory with the non-commuting 'quantum' variables of operator algebra theory, and it has led to exciting new approaches in both disciplines.
		In the next section, the operator space analogues of the nuclear, integral and absolutely summing mappings are discussed.
		In what is perhaps the deepest part of the book, the authors present the remarkable 'non-classical' phenomena that occur when one considers local reflexivity and exactness for operator spaces.
	www.oup.com /us/catalog/general/subject/Mathematics/PureMathematics/~~/c2Y9YWxsJnNzPWF1dGhvci5hc2Mmc2Q9YXNjJnBmPTQwJnZpZXc9dXNhJnByPTEwJmJvb2tDb3ZlcnM9eWVzJmNpPTAxOTg1MzQ4MjU= (345 words)

	Math 7334 Assignments (Site not responding. Last check: 2007-10-24)
		Prove that a self-adjoint operator on a finite-dimensional Hilbert space has a cyclic vector iff all of its eigenvalues are simple.
		Let P be the operator whose graph is the closure of the graph of the operator in step 1.
		Determine the spectrum of A. An operator B on a Hilbert space is said to be compact relative to a self-adjoint operator A iff D(B) contains D(A) and B(A + iI)
	www.math.gatech.edu /~harrell/7334/HW7334x.html (762 words)

	Support Operator Method Derivation: Outline (Site not responding. Last check: 2007-10-24)
		Explicitly define one of the operators (in this case, the divergence operator)
		Define the remaining operator (in this case, the gradient operator) as the discrete adjoint of the first operator
		In other words, the first operator is set up explicitly, and the second operator is defined in terms of the first operator's definition.
	www.lanl.gov /Augustus/1999_10_21/node8.html (74 words)

	Directory of operator algebraist home pages
		Operator algebraists directory at the Institute of Mathematics of the Romanian Academy, maintained by Birant Ramazan.
		Operator theory on Krein spaces, model theory for families of operators.
		Operator algebra ideas applied to the study of wavelets, statistical mechanics, representation theory, and quantum physics.
	darkwing.uoregon.edu /~ncp/OpAlgResources/HomePageDir/homepagedir.html (1056 words)

	General self adjoint operator
		as being a "cumulative eigenspace" operator: it maps vectors onto the subspace spanned by all of the "eigenvectors" whose "eigenvalues" are less than or equal to α.
		I mean, Hilbert spaces where hermitean operators are associated with random variables, and basically all that theory used in quantum mechanics.
		They say; "....the spectrum of d/dt considered as an operator on C[0, infinity], is precisely the half-plane R(L) <=0, the points on the imaginary axis except for L=0 are in the continuous spectrum and the rest of the half plane is in the point spectrum.
	www.physicsforums.com /showthread.php?t=87719 (1799 words)

	No Title (Site not responding. Last check: 2007-10-24)
		should be an arbitrary atomic measure which has the eigenvalues as its atoms.) Generally two such multiplication operators are unitarily equivalent if and only if the measures are equivalent i.e.
		The existence of a cyclic vector easily implies that the eigenvalues should be distinct.
		Then we can diagonalize it in an orthonormal basis, and the problem is in a way reduced to the case when
	mystic.math.neu.edu /courses/funcan/unit4/unit4.html (667 words)

	Adjoint Operator (Site not responding. Last check: 2007-10-24)
		Next: Adjoint Operator for Differential-Integral Up: Adjoint Equation for Radiative Previous: Adjoint Equation for Radiative
		The following is an example of how to find the adjoint operator.
		The adjoint operator for a constant c is itself, because
	asd-www.larc.nasa.gov /~yhu/paper/thesisall/node37.html (91 words)

	Fields Institute - Workshop on Nonself-adjoint Operator Algebras
		The workshop will be held in Toronto at the Fields Institute July 8-12, 2002 with the option to extend the stay either before or after to work with colleagues.
		Recent contributions to the general theory of operator algebras
		Spans and sums of unitary and similarity orbits of a single operator
	www.fields.utoronto.ca /programs/scientific/02-03/nsoa (207 words)

	Functional Analysis 2 (Site not responding. Last check: 2007-10-24)
		This course is meant to familiarize the student with sufficent knowledge of Hilbert space theory and operator theory to lead to a proof of the spectral theorem for a normal operator on a Hilbert space and some applications.
		Definition of Hilbert spaces and bounded operators on a Hilbert space, the adjoint of an operator, self-adjoint, normal, and unitary operators, orthogonal projections;
		Banach algebras, spectrum and resolvent of an element in a unital Banach algebra, the Gelfand representation theory for unital commutative Banach algebras;
	spot.colorado.edu /~packer/8340.html (233 words)

	An Algebraic Schwarz Theory (ResearchIndex) (Site not responding. Last check: 2007-10-24)
		Abstract: This report contains a collection of notes on abstract additive and multiplicative Schwarz methods for self-adjoint positive linear operator equations.
		0.1: A Polylogarithmic Bound for an Iterative Substructuring..
		An Empirical Evaluation of Dependence Analysis in Parallel Program..
	sherry.ifi.unizh.ch /95301.html (419 words)

	Wiley::A User's Guide to Operator Algebras
		Please allow an additional 3 days delivery time for paperbacks, and 10 days for hardcovers.
		The subject of operator algebras has experienced tremendous growth in recent years with significant applications to areas within algebraic mathematics as well as allied areas such as single operator theory, non-self-adjoint operator algegras, K-theory, knot theory, ergodic theory, and mathematical physics.
		This book makes recent developments in operator algebras accessible to the non-specialist.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471311359,descCd-description.html?print=true (97 words)

	DIT:: School of Mathematical Sciences: Alexander V. Kiselev (Site not responding. Last check: 2007-10-24)
		On the similarity problem for the nonselfadjoint extensions of symmetric operators, Stockholm University, Dept. of Math.
		The similarity problem for the nonselfadjoint operators with absolutely continuous spectrum: restrictions to the spectral subspaces, Stockholm University, Dept. of Math.
		Talk: Weak snnihilators for non-self-adjoint operators with almost Hermitian spectrum: International Conference "Operator Theory and Its Applications in Mathematical Physics", July 5-12 2004, Banach Center, Bedlewo, Poland.
	www.maths.kst.dit.ie /akiselev.html (511 words)

	TemplateSVL::AdjLinearOp< Scalar > class Reference
		AdjLinearOp creates the adjoint operator of an LinearOp as a linear operator in its own right.
		Note that this object points to the original linear operator.
		The primary methods are precisely those of LinearOp; in particular, the image methods such as apply() and applyAdj() work by simply calling the appropriate methods of the original linear operator.
	www.caam.rice.edu /~adpadu/svldoc/classSVL_1_1AdjLinearOp.html (127 words)

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