Exponential Stability of Operators and Operator Semigroups

Let ’ be a positive and non-decreasing function dened on the real half-line andU be a strongly continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space. We prove that if ’ andU satisfy a certain integral condition (see the relation (2) below) then U is uniformly exponentially stable. For ’ continuous, this result is due to S. Rolewicz.

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A new proof for a Rolewicz's type theorem: An evolution semigroup approach

LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {Tnx:n=0,1,...} and weak orbits {〈Tnx,x*〉:n=0,1,...} ofT wherex∈X andx*∈X*. Further we study the local capacity of operators and prove that there is a residual set of pointsx∈X with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.

Orbits, weak orbits and local capacity of operators

Exponential stability of nonlinear differential repetitive processes with applications to iterative learning control

Let R+ be the set of all non-negative real numbers, I ∈ {R,R+} and UI = {U(t, s) : t ≥ s ∈ I} be a strongly measurable and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let φ : R+ → R+ be a strictly increasing function and E be a normed function space over I satisfying some properties, see Section 2. We prove that if φ ◦ (χ[s,∞)(·)||U(·, s)x||) defines an element of the space E for every s ∈ I and all x ∈ X and if there exists M > 0 such that sup s∈I |φ ◦ (χ[s,∞)(·)||U(·, s)x||)|E = M 0 for all t > 0 and if there exists K > 0 such that

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A Theorem of Rolewicz's type in solid function spaces

We prove that if a bounded linear operator A on a Banach space X is such that, for any x ∊ X and any y ∊ X∗, the sequence 〈Ak.x,y,〉 is in; is in lp, where p ∊ (1, ∞), then the spectral radius of A is smaller than one. This solves the discrete-time version of a problem raised by Pritchard and Zabczyk (1983). As a consequence, if the linear time-invariant discrete-time systems associated with A are lq-input-bounded state stable, where q∊(1,∞), then A is power stable.

Groege Weiss

Papers

Weakly /pstable linear operators are power stable