Abstract
We shall be concerned with implemented semigroups of the form \U(t)X := T(t)XS(t) for X ∈ \L(F,E) and t ≥ 0 , where (T(t)) t ≥ 0 and (S(t)) t ≥ 0 are C 0 -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator’’ with the help of inter- and extrapolation techniques.
Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results.
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Communicated by Rainer Nagel
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Alber, J. On implemented semigroups. Semigroup Forum 63, 371–386 (2001). https://doi.org/10.1007/s002330010082
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DOI: https://doi.org/10.1007/s002330010082