Fredholm properties of evolution semigroups (Q1766867)

The authors consider questions related to the linear differential equation \[ u'(t)= A(t)u(t) \] on a Banach space \(X\) in which the operators \(A(t)\) might be unbounded. Let \(\{U(t,\tau)\}_{t\geq\tau}\), \(t,\tau\in\mathbb{R}\), denote a strongly continuous exponentially bounded evolution family on \(X\). The corresponding evolution semigroup is given by \[ (E^t u)(\tau)= U(\tau,\tau- t)u(\tau- t),\quad \tau\in\mathbb{R},\quad t\geq 0. \] They prove that the spectrum and the Fredholm spectrum for the evolution semigroup are the same. (The Fredholm spectrum of an operator \(A\) is given by \(\{X \in\mathbb{C}: \lambda- A\) is not Fredholm\}.) It is also proven through the use of the Kato lower bound that the range of \(E^t- I\) being closed and the range of \(\mathbb{G}\) being closed are equivalent properties, where \(\mathbb{G}\) denotes the evolution semigroup generator. (The Kato lower bound \(\gamma(T)\) for a closed operator \(T\) is given by \[ \gamma(T)= \inf_{\substack{ x\in\text{dom}(T)\\ Tx\neq 0}} {\| Tx\|\over \text{dist}(x,\ker T)}. \] It follows that \(\gamma(T)> 0\) if and only if the range of \(T\) is closed.) Similar results are proven for evolution semigroups acting on spaces of periodic \(X\)-valued functions and also for those acting on spaces of \(X\)-valued functions on the half-line.