Summary
For operator semigroups of class (C0) on a Banach space X it is well known that the saturation class can be characterized as the relative completion with respect to X of the domain of the infinitesimal generator. This remains true for strongly measurable semigroups {T(t),t>0} having a closed infinitesimal operator A0, but it becomes false if A0 is non-closed. We prove that a characterization is given by ∥A0T(t)f ∥=0(1), t→0 + for a fairly general class of semigroups, including certain particular semigroups which belong to Oharu's class (C(1)), or are of growth order less than one.
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The second named author was supported by a DFG grant (Go 261/4-1) which is gratefully acknowledged.
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Görlich, E., Pontzen, D. On approximation by operator semigroups of a general type. Annali di Matematica pura ed applicata 136, 119–132 (1984). https://doi.org/10.1007/BF01773380
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DOI: https://doi.org/10.1007/BF01773380