Linear operator equations

Lumer, G.; Rosenblum, Marvin

doi:10.1090/s0002-9939-1959-0104167-0

Cited by 122 publications

(69 citation statements)

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supporting

confidence: 71%

“…We also describe the essential spectra and index functions of the restrictions of these operators to norm ideals. These results thus complement the spectral analysis of multiplications initiated by Lumer and Rosenblum [20].Let % denote a separable, infinite-dimensional, complex Hilbert space and let £,(%) denote the algebra of all bounded linear operators on %. For operators A and B in £(%), let S = S (A, B) and <3" = <5(A, B) denote the operators on &(%)…”

supporting

confidence: 62%

“…. , Bn) denote commutative subsets of t(%), and define the elementary operator <& on t(%) by <3l(A") = 2"=, AiXBi [20]. Spectral properties of ÇR.…”

mentioning

confidence: 99%

“…Spectral properties of ÇR. were studied by Lumer and Rosenblum (in the more general context of Banach algebras), who proved the spectral inclusion formula°C &) C { 2 «,&: «, Ê a(Af), ßt E o(B,), 1< / < n J (#) [20,Theorem 4].…”

mentioning

confidence: 99%

“…[1], [2], [8], [10]). For these operators, the spectral inclusion (*) reduces to equality [20,Theorem 10], [9,Theorem 3.2]. Moreover, the spectral analysis of generalized derivations is essentially complete: the right and left spectra [11], the approximate point and defect spectra [7], the essential spectrum [13], the case of dense range [12], and the semi-Fredholm domain and index function [14] have all been described.…”

mentioning

confidence: 99%

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Essential spectra of elementary operators

Fialkow¹

1981

Trans. Amer. Math. Soc.

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Abstract. This paper describes the essential spectrum and index function of the operator X -» AXB, where A, B, and X are Hubert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.1. Introduction. The purpose of this note is to describe the Fredholm essential spectrum and index function for a class of operators of the form X -» AXB, where A, B, and X are Hilbert space operators. We also describe the essential spectra and index functions of the restrictions of these operators to norm ideals. These results thus complement the spectral analysis of multiplications initiated by Lumer and Rosenblum [20].Let % denote a separable, infinite-dimensional, complex Hilbert space and let £,(%) denote the algebra of all bounded linear operators on %. For operators A and B in £(%), let S = S (A, B) and <3" = <5(A, B) denote the operators on &(%)

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supporting

confidence: 71%

supporting

confidence: 62%

“…. , Bn) denote commutative subsets of t(%), and define the elementary operator <& on t(%) by <3l(A") = 2"=, AiXBi [20]. Spectral properties of ÇR.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Essential spectra of elementary operators

Fialkow¹

1981

Trans. Amer. Math. Soc.

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Backstepping‐based output regulation for systems with infinite‐dimensional actuator and sensor dynamics

Kerschbaum

Deutscher

2016

Proc Appl Math and Mech

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The output regulation problem is solved for an ODE plant driven by a parabolic actuator with spatially varying coefficients. Applying the backstepping transformation to the actuator allows the solution of the regulator equations in closed‐form and thus the formulation of the solvability condition. An output feedback regulator is obtained by designing a disturbance observer using delayed measurement and a reference observer that both achieve finite‐time convergence. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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On formulas in closed form for Van Vleck expansions

Gora

1972

Int J of Quantum Chemistry

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AbstractsCanonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vleck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. I t will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. 'The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two-to .four-dimensional problems.Des transformations canoniques ont Ct C employtes frkquemment pour simplifier 1'Hamiltonien et d'autres optrateurs. Dans les thtories moltculaire et de l'ttat solide le dCveloppement de Van Vleck est gtneralement utilisC B cette fin, tandis que dans les thtories des particules intbragissant avec des champs on a prtfCrt une combinaison de transformations canoniques dans une forme close avec un type de dtveloppement de Van Vleck. Pour quelques-unes des transformations employkes dans les applications des formules en forme close sont bien connues. I1 est dtmontrC ici, que de telles formules peuvent Ctre obtenues quand la fonction de transformation est bilintaire dans les variables canoniques, et aussi qu'B l'aide des optrateurs matriciels il est possible de simplifier considCrablement les dkductions correspondantes. Avec le thCoreme de Cayley-Hamilton on peut exprimer les dtveloppements pour les optrateurs matriciels dans une forme close. Le nombre de termes d'optrateurs distincts ainsi obtenu est le mCme que le rang des matrices utilistes. Pour calculer les coefficients de ces termes d'opCrateurs on introduit un nouveau type de fonctions spkciales. Les transformations canoniques lintaires, qui en rCsultent comprennent des rotations gtntralistes dans l'espace ordinaire et dans l'espace des phases. Des resultats explicites ont t t t obtenus pour plusieurs problemes deux jusqu'8 quatre dimensions.Kanonische Transformationen sind oft verwendet worden um Hamilton-und andere Operatoren zu vereinfachen. In Molekul-und Festkorpertheorie wird gewohnlich die SOgenannte Van Vleck-Entwicklung zu diesem Zweck angewandt, wahrend in Theorien

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Linear operator equations

Cited by 122 publications

References 2 publications

Essential spectra of elementary operators

Essential spectra of elementary operators

Backstepping‐based output regulation for systems with infinite‐dimensional actuator and sensor dynamics

On formulas in closed form for Van Vleck expansions

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