Frechet differentiability of parameter-dependent analytic semigroups

We study the dependence on a vector-valued parameter q of a collection of a nalytic semigroups {T(t;q), t greater than or equal to 0} arising, for exam ple, from a collection of diffusion-convection equations whose infinitesima l generators are abstract elliptic operators defined in terms of sesquiline ar forms in a "Gelfand triple" or "pivot space" framework. Within a mathema tical framework slightly more general than the one set forth below, Banks a nd Ito [Banks, H. T. and Ito, K., "A unified framework for approximation in inverse problems for distributed parameter systems," Control-Theory and Ad vanced Technology, 4 (1988), pp. 73-90] have shown, as an application of th e Trotter-Kato Theorem, that the map q H T(t; q) is continuous in the stron g operator topology. In this paper, we establish the analyticity of this ma p in the uniform operator topology, and exhibit its Frechet derivative both as a contour integral and as the solution of a particular initial-value pr oblem. (C) 1999 Academic Press.