Fr´echet Differentiability of Parameter-Dependent Analytic Semigroups

We study the dependence on a vector-valued parameter q of a collection of analytic semigroups T t; q., tG04 arising, for example, from a collection of diffusion-convection equations whose infinitesimal generators are abstract elliptic operators defined in terms of sesquilinear forms in a ‘‘Gelfand triple’’ or ‘‘pivot space’’ framework. Within a mathematical framework slightly more general than the one set forth below, Banks and Ito wBanks, H. T. and Ito, K., ‘‘A unified framework for approximation in inverse problems for distributed parameter sys- tems,’’ Control}Theory and Advanced Technology, 4 1988., pp. 73]90x have shown, as an application of the Trotter-Kato Theorem, that the map q¬T t; q.is continuous in the strong operator topology. In this paper, we establish the analyticity of this map in the uniform operator topology, and exhibit its Fr´echet derivative both as a contour integral and as the solution of a particular initial-value problem.