In this thesis we study perturbation results for the generator A of a positive C_0--semigroup on a Banach lattice X. We generalize results that hold for an AM- or AL-space X and extend them to an arbitrary Banach lattice X in case the perturbation can be factorized as P=BC via an AM- or AL-space U. This allows to deal with applications on (infinite dimensional) reflexive Banach spaces, which are a priori excluded if X is an AM- or an AL-space. First, B and C are assumed to be positive operators satisfying the spectral condition r(CR(\lambda,A_{-1})B)<1 for some \lambda>\omega_0(A). We then prove that the generation results obtained in this case still hold when B and C are not necessarily positive, but only dominated by positive operators. These abstract results are applied to domain perturbations of generators, a heat equation with boundary feedback and perturbations of the first derivative. Combining the previous results we then deal with "doubly unbounded" perturbations P:Z\to X_{-1}, provided they can be factorized as P=BC via U=\mathbb{R}^N. In this context, we prove a Staffans--Weiss type perturbation result. We show that the classically assumed admissibility and invertibility conditions for the associated input--output map \mathcal{F}_\infty follow from the (much simpler to verify) spectral condition mentioned above. The abstract results are applied to domain perturbations and finite rank perturbations of the first derivative.

On Structured Perturbations of Positive Semigroups / Barbieri, Alessio. - (2025 Feb 20).

On Structured Perturbations of Positive Semigroups

BARBIERI, ALESSIO
2025-02-20

Abstract

In this thesis we study perturbation results for the generator A of a positive C_0--semigroup on a Banach lattice X. We generalize results that hold for an AM- or AL-space X and extend them to an arbitrary Banach lattice X in case the perturbation can be factorized as P=BC via an AM- or AL-space U. This allows to deal with applications on (infinite dimensional) reflexive Banach spaces, which are a priori excluded if X is an AM- or an AL-space. First, B and C are assumed to be positive operators satisfying the spectral condition r(CR(\lambda,A_{-1})B)<1 for some \lambda>\omega_0(A). We then prove that the generation results obtained in this case still hold when B and C are not necessarily positive, but only dominated by positive operators. These abstract results are applied to domain perturbations of generators, a heat equation with boundary feedback and perturbations of the first derivative. Combining the previous results we then deal with "doubly unbounded" perturbations P:Z\to X_{-1}, provided they can be factorized as P=BC via U=\mathbb{R}^N. In this context, we prove a Staffans--Weiss type perturbation result. We show that the classically assumed admissibility and invertibility conditions for the associated input--output map \mathcal{F}_\infty follow from the (much simpler to verify) spectral condition mentioned above. The abstract results are applied to domain perturbations and finite rank perturbations of the first derivative.
20-feb-2025
On Structured Perturbations of Positive Semigroups / Barbieri, Alessio. - (2025 Feb 20).
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Descrizione: On Structured Perturbations of Positive Semigroups
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/260600
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