Spectral inclusions for semigroups of closed operators (Q1971724)

The paper under review is concerned with one-parameter semigroups of closed operators and the spectrum of their generators. Let \(X\) denote a complex Banach space and \((T_t)_{t> 0}\) closed operators with \(T_t T_s\subset T_{t+s}\) with a non-trivial subspace \(D\) such that \(t\mapsto T_i\) is continuous on \(D\). Assume the existence of a regularizing operator \(C\), i.e., in \(C\subseteq D\) such that \(C^{-1} T_tC= T_t\) for all \(t\geq 0\). The ``infinitesimal generator'' is replaced by the following construction: Let \(A_0:= \{(x,y)\in D: (T_t x-x)/t\to y\}\) and \(A= \{(x,y): (Cx, Cy)\in A_0\}\). The regularized semigroup \((S_t= T_tC)_{t> 0}\) and the resolvent and spectrum defined by \(C\) are the main tools for investigation of the spectral behaviour of a class (unbounded) closed operators, a class which is wide enough to cover important examples as integrated semigroups, distribution semigroups and some other semigroups which are continuous for \(t>0\) and injective. The paper is closed with examples and discussions clearifying some imprecise statements in the literature.