Growth of orbits for operator semigroups on Banach spaces with Schauder bases.

Let X be a Banach space with a normalized (Schauder) basis (b k) k and let { T (t) } t ≥ 0 be a C 0 -semigroup on X with generator A : D (A) ⊂ X → X . The main contribution of this paper is a new estimate for the semigroup orbits of initial data in D (A 2) that have with respect to (b k) k "absolutely summable graph norm." This result is noteworthy because (1) there is no requirement that z ↦ R (z , A) x has a bounded analytic extension to C + [e.g. Van Neerven (Semigroup Forum 53(2):155–161, 1996)] and (2) the only structural condition on X is the existence of (b k) k [e.g. Wrobel (Indiana Univ. Math. J. 38(1):101–114, 1989)]. Two directions for additional research are in some detail discussed as well at the end of this paper. [ABSTRACT FROM AUTHOR]