Abstract.
In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence \(A_{d} ^{n} ,\,n \in \mathbb{N},\) where A d = (I + A) (I − A)−1.
We show that if A and A −1 generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A d is equivalent to the uniform boundedness of the semigroup generated by A.
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Guo, B.Z., Zwart, H. On the Relation between Stability of Continuous- and Discrete-Time Evolution Equations via the Cayley Transform. Integr. equ. oper. theory 54, 349–383 (2006). https://doi.org/10.1007/s00020-003-1350-9
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DOI: https://doi.org/10.1007/s00020-003-1350-9