In this book, non-spectral methods are presented and discussed that have been developed over the last two decades for the investigation of asymptotic behavior of operator semigroups. This concerns in particular Markov semigroups in L¹-spaces, motivated by applications to probability theory and dynamical systems. Recently many results on the asymptotic behaviour of Markov semigroups were extended to positive semigroups in Banach lattices with order-continuous norm, and to positive semigroups in non-commutative L¹-spaces. Related results, historical notes, exercises, and open problems accompany each chapter.

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In this book, non-spectral methods are presented and discussed that have been developed over the last two decades for the investigation of asymptotic behavior of one-parameter operator semigroups in Banach spaces. This concerns in particular Markov semigroups in L¹-spaces, motivated by applications to probability theory and dynamical systems. Recently many results on the asymptotic behaviour of Markov semigroups were extended to positive semigroups in Banach lattices with order-continuous norm, and to positive semigroups in non-commutative L¹-spaces. Related results, historical notes, exercises, and open problems accompany each chapter.

The book is directed to graduate students and researchers working in operator theory, particularly those interested in C₀-semigroups in classical and non-commutative L¹-spaces, in mean ergodic theory, and in dynamical systems.

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