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We propose an abstract framework for the computation of the spectrum (A) of a linear operator A : D(A) X ! X on a Banach space X through a condition in a smaller Banach space X 1. If this space is nite dimensional this yields a characteristic equation for (A). The method is tested for delay, integro-diierential and population equations and is applicable to… (More)
In this article we survey results concerning asymptotic properties of C0-semigroups on Banach spaces with respect to the weak operator topology. The property " no eigenvalues of the generator on the imaginary axis " is equivalent to weak stability for most time values; a phenomenon called " almost weak stability ". Further, sufficient conditions implying… (More)
We give conditions on a strongly continuous semigroup T and a bounded operator B, such that the perturbed semigroup inherits the regularity properties of T. 1. Introduction The aim of this paper is to understand why some regularity properties of strongly continuous semigroups persist under bounded perturbation while others do not. This is a very old… (More)
For a strongly continuous semigroup (T (t)) t≥0 with generator A we introduce its critical spectrum σ crit (T (t)). This yields in an optimal way the spectral mapping theorem σ(T (t)) = e tσ(A) ∪ σ crit (T (t)) and improves classical stability results.
In this paper, we present two quite general approximation theorems for the propagators of higher order (in time) abstract Cauchy problems, which extend largely the classical Trotter-Kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions. Then, we apply the approximation theorems to deal with the second… (More)
We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0-semigroup on C 0 (R − ,X) yielding the solutions of the equation.
For a generator A of a C 0-semigroup T (·) on a Banach space X we consider the semi-norm M k x := lim sup t→0+ t −1 (T (t) − I)A k−1 x on the Favard space F k of order k associated with A. The use of this semi-norm is motivated by the functional analytic treatment of time-discretization methods of linear evolution equations. We show that sharp inequalities… (More)