arXiv subjects

Martin Adler

Publications and source records attributed to Martin Adler.

4 recordsLinked to original sources

arXiv2016

Spectral theory for structured perturbations of linear operators

We characterize the spectrum (and its parts) of operators which can be represented as G=A+BC for a simpler operator A and a structured perturbation BC. The interest in this kind of perturbations is motivated, e.g., by perturbations of the domain of an operator A but also arises in the theory of closed-loop systems in control theory. In many cases our results yield the spectral values of G as zeros of a "characteristic equation".

math.SP↗

arXiv2015

A semigroup approach to the numerical range of operators on Banach spaces

We introduce the numerical spectrum $σ_n(A)\subset \mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $σ_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups.

math.FA↗

arXiv2015

Asymptotic properties of $C_0$-semigroups under perturbations

For a given $C_0$-semigroup $(T(t))_{t\geq 0}$ we consider Staffans-Weiss perturbations $(B,C)$ of its generator as studied in [1] and investigate the robustness of asymptotic properties of the perturbed $C_0$-semigroup $(T_{BC}(t))_{t\geq 0}$. As a concrete application we study the asymptotic behavior of a neutral semigroup.

math.FA↗

arXiv2015

Perturbation of Analytic Semigroups and Applications to Partial Differential Equations

In a recent paper we presented a general perturbation result for generators of $C_0$-semigroups. The aim of the present paper is to replace, in case the unperturbed semigroup is analytic, the various conditions appearing in this result by simpler assumptions on the domain and range of the operators involved. The power of our result to treat classes of PDE's systematically is illustrated by considering a generic example, a degenerate differential operator with generalized Wentzell boundary conditions and a reaction diffusion equation subject to Neumann boundary conditions with distributed unbounded delay.

math.FA↗