Existence families, functional calculi and evolution equations
The subject of the book under review is the theory of regularized semigroups. The notion consists in a generalization of \(C_ 0\)- semigroups and its aim is to solve linear Cauchy problems. More precisely, let \(C\) be a bounded injective operator on a Banach space \(X\). A \(C\)-regularized semigroup is a strongly continuous mapping \(W: [0,\infty)\to {\mathcal L} (X)\) such that \[ W(0)=C, \quad CW(t+s)= W(t) W(s) \qquad (t,s\geq 0). \] Thus, for \(C\) the identity we find the usual notion of a \(C_ 0\)-semigroup. Similar to this classical case one can define the infinitesimal generator \(A\) of a \(C\)-semigroup \(W\). It is a closed linear operator which commutes with \(C\). Moreover, for all \(x\) in the image of \(C\), the Cauchy problem \[ \begin{cases} u'(t)= Au(t) \qquad (t\geq 0)\\ u(0)= x\end{cases} \tag{CP} \] has a unique mild solution (given by \(u(t)= W(t)x\)). Regularized semigroups were introduced by Da Prato (1966). The interest in the subject arose again recently and related concepts were introduced. These lecture notes are based on a series of articles of the author from 1987 to 1994. Here is a sample of subjects treated in the book: Diverse notions of existence and uniqueness, Hille-Yosida theorems, holomorphic regularized semigroups, integrated semigroups, functional calculus, reversibility of solutions, boundary of a holomorphic semigroup, systems, non-autonomous equations. Applications concern the Schrödinger equation, correct systems in the sense of Petrovsky, equipartition of energy. The aim of the book is not to present a development which parallels the theory of \(C_ 0\)-semigroups. The intention of the author is rather to describe a variety of new phenomena which can occur for regularized semigroups. He takes care to explain the diverse notions and to illustrate them by examples and counterexamples.
correct systems in the sense of Petrovsky
scientific article; zbMATH DE number 556114