Abstract
Let X be a Banach space with a normalized (Schauder) basis \((b_{k})_{k}\) and let \(\{T(t)\}_{t\ge 0}\) be a \(C_{0}\)-semigroup on X with generator \(A:D(A)\subset X\rightarrow X\). 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 \(z\mapsto R(z,A)x\) has a bounded analytic extension to \({\mathbb {C}}_{+}\) [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.
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Communicated by Abdelaziz Rhandi.
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Gaebler, H. Growth of orbits for operator semigroups on Banach spaces with Schauder bases. Semigroup Forum 105, 426–433 (2022). https://doi.org/10.1007/s00233-022-10312-3
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DOI: https://doi.org/10.1007/s00233-022-10312-3