Topics in operator semigroups
This (graduate level) volume deals with the classical linear theory of strongly continuous semigroups and with various developments (most of recent vintage) related with this subject. The classical theory (generators, the Hille-Yosida theorem, dissipative operators, the Trotter convergence theorem, exponential formulas, perturbations, groups, Stone's theorem, analytic semigroups) is dealt with in Part I, ``General theory''. Among less standard subjects in this part are the following. (a) Semi-simplicity spaces for groups, a Banach space version of the spectral integral representation of bounded strongly continuous groups in Hilbert spaces (the Hilbert space result is a consequence of Stone's theorem plus Sz.-Nagy's similarity characterization of bounded strongly continuous semigroups). (b) Semigroups as functions of their generators, including a noncommutative Taylor formula and semigroups whose generators depend analytically on a parameter. (c) Boundary values of semigroups analytic in a half-plane, with applications to the Gauss-Weierstrass and Cauchy-Poisson semigroups and the Gamma semigroup. (d) Pre-semigroups (otherwise known as regularized semigroups) and their relation to the abstract Cauchy problem. There is also a discussion of stability, mean stability and related questions. Part II, ``Integral representations'', is related to the semi-simplicity spaces in Part I, but it deals with integral representations of more general families of operators such as semigroups of unbounded operators and families of unbounded symmetric operators (among them local symmetric semigroups and local symmetric cosine functions). Part III, ``A taste of applications'', has two subparts. In the first, the subject is a linear evolution equation \(u'(t, z) = A(t, z)u(t, z)\), \(u(0, z) = u\), and the emphasis is on establishing analyticity in \(z\) of the evolution operator from analyticity of \(A(t, z)\) or of its resolvent. This is shown under two different sets of conditions on \(A(t, z)\) that allow construction of the evolution operator (one set of conditions due to Kato, the other due to Tanabe). The second subpart deals with similarity (two bounded or unbounded operators \(A,\) \(B\) are similar if \(B = Q^{-1}AQ\) for bounded invertible \(Q)\). Mostly, similarity is considered within the family \(S + \zeta V\), where \(iS\) generates a strongly continuous group and the bounded operator \(V\) satisfies the Volterra relations \(VD(S) \subset D(S)\) and \(SV - VS \subset V^2\). The exposition is evenly paced and essentially self-contained.
scientific article; zbMATH DE number 5625335