Generators of operator semigroups (Q1005465)

The theory of operator semigroups is experiencing a renewed and increasing interest in the research community motivated by the applicability of the theory to control problems or stochastic differential equations, just to mention a couple of branches of actual interest. An operator semigroup is a linear dynamical system \(T:\mathbb{R}_+\to {\mathcal L}(X)\) in a Banach space \(X\) satisfying \(T(t+s)=T(t)T(s)\) and suitable continuity properties, associated to an abstract differential equation. For example, the most known class of \(C_0\)-semigroups consists of the semigroups that are continuous in the strong operator topology, i.e., the map \(t\mapsto T(t)x\) is continuous for \(t\geq 0\) and for every \(x\in X\). In the paper under review the author contributes to the spectral theory of the so-called (E)-semigroups in the sense of Hille-Phillips, which are strongly continuous for \(t>0\) and satisfy a weaker ergodic condition at \(t=0\). To build up a theory analogous to that of \(C_0\)-semigroups, the author has to work with linear relations instead of operators. The main technical tool is the so-called leading infinitesimal generator, which is the linear relation defined by \[ \mathbb{A}:=\left\{(x,y):x\in \overline{\operatorname{im} T},\;T(t)x-T(s)x = \int_s^t T(r)y \,dr\right\}. \] The main difference from the classical theory is that the vector \(y\) is not necessarily uniquely defined, allowing one to define other notions generalizing the concept of a generator. Developing some technical tools, the author obtains results concerning the spectral properties of generators and decomposition theorems.