Subordinated discrete semigroups of operators

Dungey, Nick

doi:10.1090/S0002-9947-2010-05094-9

Subordinated discrete semigroups of operators
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by Nick Dungey

Trans. Amer. Math. Soc. 363 (2011), 1721-1741

DOI: https://doi.org/10.1090/S0002-9947-2010-05094-9

Published electronically: November 15, 2010

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Abstract:

Given a power-bounded linear operator $T$ in a Banach space and a probability $F$ on the non-negative integers, one can form a ‘subordinated’ operator $S = \sum _{k\geq 0} F(k) T^k$. We obtain asymptotic properties of the subordinated discrete semigroup $(S^n\colon n=1,2, \ldots )$ under certain conditions on $F$. In particular, we study probabilities $F$ with the property that $S$ satisfies the Ritt resolvent condition whenever $T$ is power-bounded. Examples and counterexamples of this property are discussed. The hypothesis of power-boundedness of $T$ can sometimes be replaced by the weaker Kreiss resolvent condition.

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