Operator Multipliers Generating Strongly Continuous Semigroups

Graser, Thomas J.

doi:10.1007/pl00005912

Cited by 8 publications

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“…Then r → e tA(r) f (r) is continuous, too. We have therefore shown that the multiplication semigroups e tA(·) and e tB(·) , generated by the multiplication operators A(·) and B(·), both act on the space X = C 0 (R; X), see also Graser [12]. It can be seen by induction that…”

Section: A Product Formulamentioning

confidence: 91%

“…We denote the evolution family solving (NCP) by W and the corresponding evolution semigroup, generated by the closureC of C := − d ds + A(·) + B(·), by W. As we shall see in a moment, Assumption 2.1 yields that the multiplication operators A(·), B(·) with appropriate domain generate strongly continuous multiplication semigroups on C 0 (R; X) (for more on this matter we refer to Engel and Nagel [8, Sec. III.4.13] and Graser [12]). for all x ∈ X, locally uniformly in s, t with s ≤ t. .…”

Section: A Product Formulamentioning

confidence: 99%

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Operator splitting for non-autonomous evolution equations

Bátkai

Csomós

Farkas

et al. 2011

Journal of Functional Analysis

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We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous problems. The results are then illustrated by the example of a imaginary time Schr\"odinger equation with time dependent potential. We also obtain convergence rates for the Strang-splitting applied to this problem.Comment: Referees suggestions incorporated, misprints corrected. Final versio

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Section: A Product Formulamentioning

confidence: 91%

Section: A Product Formulamentioning

confidence: 99%

Operator splitting for non-autonomous evolution equations

Bátkai

Csomós

Farkas

et al. 2011

Journal of Functional Analysis

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“…[4]). In particular, for every scalar function φ ∈ C b (IR) we obtain a multiplication operator M φ by…”

Section: We Denote Its Generator By (G D(g))mentioning

confidence: 99%

Evolution semigroups for nonautonomous Cauchy problems

Nickel

1997

Abstract and Applied Analysis

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Abstract. In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problemsa Banach space X by the existence of certain evolution semigroups.Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some "concrete" conditions. As a typical example, we discuss the so called "parabolic" case. Basic definitionsIn this section, we introduce the basic definitions and notations in order to discuss nonautonomous Cauchy problems in terms of evolution families and evolution semigroups. In addition, we mention some of their fundamental properties.The solution of a nonautonomous Cauchy problem on some Banach space X can be given by a so called evolution family which can be defined as follows. Definition 1.1 (Evolution family). A family (U (t, s)) t≥s of linear, bounded operators on a Banach space X is called an (exponentially bounded) evolution family iffor some M ≥ 1, ω ∈ IR and all t ≥ s ∈ IR.1991 Mathematics Subject Classification. Primary 47D06; secondary 47A55.

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“…Such operators (and semigroups generated by them) have been studied by, e.g. Holderrieth [12] as well as Hardt and Wagenführer [11] for matrix multiplication semigroups and by Arendt and Thomaschewski [2] and Graser [10] in the infinite dimensional case. See also [14,Section 4] and [17].…”

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confidence: 99%

Eigenvalues and stability properties of multiplication operators and multiplication semigroups

Heymann

2013

Mathematische Nachrichten

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Abstract. We investigate uniform, strong, weak and almost weak stability of multiplication semigroups on Banach space valued L p -spaces. We show that, under certain conditions, these properties can be characterized by analogous ones of the pointwise semigroups. Using techniques from selector theory, we prove a spectral mapping theorem for the point spectra of the pointwise and global semigroups and apply this as a major tool for determining almost weak stability.One of the significant features of the Fourier transform is that it converts a differential operator into a multiplication operator induced by some scalar-valued function. The properties of the original operator are then determined by the values of this function. The same holds if a system of differential operators is transformed into a matrix valued multiplication operator on a vector valued function space. This motivates the systematic investigation of multiplication operators on Banach space valued function spaces.Such operators (and semigroups generated by them) have been studied by, e.g. (see e.g. [4]). In fact, motivated by these applications, Hans Zwart has proved a characterisation of strong stability of a multiplication semigroup in the finite dimensional case (see [18]), while so-called polynomial stability of multiplication semigroups is characterized in Theorem 4.4 of [3].The general question in this context is to what extent the global properties of a multiplication operator are determined by the local properties of the pointwise operators. In this paper we systematically investigate spectral and stability properties of multiplication semigroups. Our aim is thus to understand how these properties are related to those of the corresponding pointwise operators or semigroups, as explained below. As a major tool and also as a result of independent interest, we obtain a perfect characterisation of the eigenvalues of multiplication operators (Theorem 6). Furthermore, we indicate how the global and local stability properties 1991 Mathematics Subject Classification. 47D06.

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Operator Multipliers Generating Strongly Continuous Semigroups

Cited by 8 publications

References 0 publications

Operator splitting for non-autonomous evolution equations

Operator splitting for non-autonomous evolution equations

Evolution semigroups for nonautonomous Cauchy problems

Eigenvalues and stability properties of multiplication operators and multiplication semigroups

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