On weak decay rates and uniform stability of bounded linear operators

Glück, Jochen

doi:10.1007/s00013-015-0746-5

Cited by 5 publications

(7 citation statements)

References 13 publications

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“…The conditions for uniform stability in Proposition 4.2 can actually be further weakened: it suffices if we replace either the strong convergence that takes place with respect to a rate in assertion (ii), the p-integrability of the orbits in assertion (iii), or the summability condition in assertion (iv), with corresponding weak properties-where weak means that we test against functionals. Results of this type can, for instance, be found in [19,42,52]. Thus, one also obtains a characterisation of r (T ) < 1 in terms of certain weak stability properties of the operator T .…”

Section: Proposition 42 Let X Be a Banach Space And Let T ∈ L(x ) The Following Assertions Are Equivalentmentioning

confidence: 80%

Stability criteria for positive linear discrete-time systems

Glück

Mironchenko

2021

Positivity

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We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.

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Section: Proposition 42 Let X Be a Banach Space And Let T ∈ L(x ) The Following Assertions Are Equivalentmentioning

confidence: 80%

Stability criteria for positive linear discrete-time systems

Glück

Mironchenko

2021

Positivity

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“…Such a result might not come as a complete surprise and it is motivated by the following observation: let T be a continuous linear operator on a, say complex, Banach space E. If, for all x ∈ E and all x ′ ∈ E ′ , the sequence ( x ′ , T n x ) n∈N0 converges to 0 with a certain rate, then the powers T n actually converge to 0 with respect to the operator norm; results of this type can for instance be found in [57], [56] and [26]. Here, we consider an operator T on a complex Banach lattice E such that for all x, x ′ ≥ 0 the sequence (d + ( x ′ , T n x ) n∈N0 has a certain decay rate.…”

Section: The Spectral Radius IImentioning

confidence: 99%

“…Then r(T ) ∈ σ(T ). For the proof of Theorem 5.2 we employ techniques from [26]. We first need to introduce a bit of terminology.…”

Section: Then R(t ) ∈ σ(T )mentioning

confidence: 99%

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Towards a Perron–Frobenius theory for eventually positive operators

Glück

2017

Journal of Mathematical Analysis and Applications

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This article is a contribution to the spectral theory of so-called eventually positive operators, i.e. operators T which may not be positive but whose powers T n become positive for large enough n. While the spectral theory of such operators is well understood in finite dimensions, the infinite dimensional case has received much less attention in the literature.We show that several sensible notions of "eventual positivity" can be defined in the infinite dimensional setting, and in contrast to the finite dimensional case those notions do not in general coincide. We then prove a variety of typical Perron-Frobenius type results: we show that the spectral radius of an eventually positive operator is contained in the spectrum; we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector; and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions. All our results are formulated for operators on Banach lattices, and many of them do not impose any compactness assumptions on the operator.

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“…then r(T ) < 1; see [7] for updated results of this type. For comprehensive information on this subject we refer the reader to [12].…”

Section: Notations Definitions and Statementmentioning

confidence: 96%