| Summary: | <p>This thesis is concerned with the quantified asymptotic theory of operator semigroups and its applications. There are two sides of this story that are studied in the two main parts of the thesis respectively. The first side of the story is the asymptotic theory of strongly continuous operator semigroups arising from the study of damped wave equations and the countable spectrum theorem for strong stability. In the first half of the thesis, we apply known theory to two coupled wave-heat systems in order to derive energy decay estimates, before developing new direct integral theory so that one may explore asymptotic ideas there. The second side of the story is the asymptotic theory of discrete operator semigroups arising from the much celebrated Katznelson--Tzafriri and mean ergodic theorems. In the second half of the thesis, we establish new abstract results in this area, proving sharp estimates for the rate of decay between consecutive powers of operators on Hilbert spaces. Additionally, we prove upper and lower bounds for this same decay in the case of multiplication-like operators that improve on previously known theory.</p>
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