On uniform exponential stability of exponentially bounded evolution families
The authors deduce new characterizations for the uniform exponential stability of evolution families in terms of rearrangement Banach function spaces. By applying their methods, the authors obtain stability conditions both for semigroups and for periodic evolution families. Let \(X\) be a Banach space, let \(J \in \{{\mathbb R}, {\mathbb R}_+\}\) and \(\Delta_J := \{(t, s) \in J \times J : t\geq s\}\). One of the main results of the paper states that if \(U=\{U(t, s) : (t, s) \in \Delta_J\}\) is an exponentially bounded evolution family on \(X\) and \(E(J)\) is a rearrangement invariant Banach function space over \(J\) such that \(\lim_{r \to \infty} \|\chi_{[0, r]}(\cdot)\|_{E^\prime(J)} = \infty,\) then the following statements are equivalent: {\parindent7mm \begin{itemize}\item[(i)] the family \(U\) is uniformly exponentially stable; \item[(ii)] there exists a positive constant \(M^\prime(J)\) such that, for each \((s, t) \in \Delta_J\) and each \(x^* \in X^*\), the map \(\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|\) defines an element of the space \(E^\prime(J)\) and \[ \sup_{(t, s) \in \Delta_J}\|\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|\;\|_{E^\prime(J)} \leq M^\prime(J)\;\|x^*\|. \] \end{itemize}}
evolution families of bounded linear operators
scientific article; zbMATH DE number 5314951