Abstract
In this paper, necessary and sufficient conditions are given for U * μ*n to converge uniformly on the real axis; here $\mu$ is a nonsingular probability measure on ℝ, and U is a Banach space valued L∞-function. A connection to uniform convergence of Cesaro mean values is shown. By applying the results to extended orbits of bounded C_0-semigroups on a Banach space X one can relate both kernel and range of the respective generator with those of the derivative operator on L∞(X). Ergodic theorems and consequences for subordinated semigroups, in particular for holomorphic semigroups, are deduced.
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Schmalmack, M. Asymptotic Behaviour of Convolution Semigroups. Semigroup Forum 71, 265–288 (2005). https://doi.org/10.1007/s00233-005-0520-2
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DOI: https://doi.org/10.1007/s00233-005-0520-2