A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory (Q882618)

Let \(T\) be the generator of a \(C_{0}\)-semigroup \((U(t))_{t \geq 0}\) on a Hilbert space \(H\). Let \(K\) be a bounded operator in \(H\) and \((V(t))_{t \geq 0}\) the \(C_{0}\)-semigroup generated by \(T+K\). \(V(t)\) is given by the Dyson-Phillips expansion \[ V(t)= \sum_{j=0}^{\infty} U_{j}(t). \] The stability of the essential and essential type spectrum is implied by the compactness of \(U_{1}(t)\) and \(U_{k}(t)\) for \(k \geq 1\), respectively. The stability of the critical and critical type spectrum is implied by the norm continuity of \(U_{1}(t)\) and \(U_{k}(t)\) for \(k \geq 1\), respectively. The author proves conditions for these properties to hold involving the resolvent of \(T\). Applications are given to transport theory, where \(T\) is the streaming operator \[ T: L^\infty(\Omega \times V) \supset D(T) \ni \varphi \mapsto -v\frac{\partial \varphi}{\partial x} - \sigma(x,v)\varphi(x,v), \] where \(\Omega\) is a domain in \({\mathbb R}^N\), \(V\) the space of velocities and \(D(T)\) involves some boundary conditions. The operator \(K\) is taken to be a collision operator of the form \[ K(x): L^2(V) \ni \varphi \mapsto \int_{V} k(x,v,v') \varphi(v') \,d\mu(v') \] which satisfies the assumption that \(\{K(x); x \in \Omega\}\) is collectively compact in \(L^2(V)\).