Entire solutions of the abstract Cauchy problem (Q1174274)

In the past several generalizations of strongly continuous one parameter semigroups and their infinitesimal generators were introduced in connection with Cauchy-problems, e.g. polynomials of generators, integrated semigroups, operator matrices and \(C\)- semigroups. A \(C\)- semigroup, \(C\) a bounded injective linear operator on a Banach space, is a continuous family \((W(t),\;t\geq 0)\), such that \(W(0)=C\) and fulfilling the functional equation \(W(t)W(s)=C\cdot W(t+s)\). The author defines in a similar way \(C\)-groups and entire \(C\)-groups in order to obtain a new and unified approach to Cauchy problems. The first chapters are concerned with definitions, general properties and examples of entire \(C\)-groups and their generators. Chapter III is concerned with first order problems and \(C\)-groups defined by polynomials of generators of holomorphic semigroups. In Chapter IV the author considers second order problems, matrices of operators and representation of the solutions by \(C\)-groups. In Chapter V models for elastic systems with damping are treated and Chapter VI is concerned with examples (heat equation, Laplace equation) which can be treated in this unified way.