One-parameter Semigroups of Positive Operators
Description
This volume develops the structural theory of one-parameter semigroups of positive operators on ordered Banach spaces, addressing the fundamental problem of characterizing their generators, spectral properties, and asymptotic behavior. Such semigroups form a central analytical framework for models arising in partial differential equations, probability theory, ergodic theory, and mathematical physics.
Since its first publication in 1986, the book has become a foundational reference, offering a rigorous and unified treatment of positivity, Banach lattices, and semigroup theory. The exposition covers Banach spaces, Banach lattices, C(X) spaces, and operator algebras, with a systematic focus on generator theory, spectral analysis, and long-time behavior.
This second edition preserves the original conceptual framework and organization while presenting the entire text in a fully revised and professionally typeset form. Misprints have been corrected, references updated, and a new chapter of notes surveys key developments of the past four decades, placing the original results in a modern context without compromising their historical coherence. The book remains an essential resource for researchers and advanced graduate students in operator theory and functional analysis.
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Persons
Wolfgang Arendt is a professor emeritus of mathematics at Ulm University. He received his doctorate from the University of Tübingen in 1979 under H. H. Schaefer and completed his habilitation in 1985. His research focuses on functional analysis, operator semigroups, and partial differential equations. He has supervised numerous doctoral students and contributed extensively to the development of the theory of evolution equations.
Annette Räbiger (née Grabosch) received her Ph.D. in mathematics from the University of Tübingen in 1985 under Rainer Nagel, working on semigroups for abstract delay equations. She later worked in mathematical biology and has been involved in mathematics education, psychology, and pedagogy.
Günther Greiner is a professor emeritus of computer science at the University of Erlangen-Nürnberg. He studied mathematics and physics at the University of Tübingen, where he earned his Ph.D. and habilitation in mathematics. He later moved into computer science, focusing on computer graphics, geometric modeling, and visual computing, and contributed to research at the interface of mathematics and computer science.
Ulrich Groh is an adjunct professor of mathematics at the University of Tübingen. He received his Ph.D. under H. H. Schaefer. After a one-year research fellowship at Hokkaido University, Sapporo, he completed his habilitation in 1984. His research focuses on the spectral theory of positive operators on C- and W-algebras. From 1985, he worked at IBM Germany, eventually heading the Education and Research division.
Ulrich Moustakas received his Dr. rer. nat. in mathematics from the University of Tübingen in 1985 with work on positive operators on Banach lattices. He later turned to theology and habilitated in systematic theology at the University of Tübingen. His work focuses on theology in the context of philosophy of science and hermeneutics.
Rainer Nagel is a professor emeritus of mathematics at the University of Tübingen and a leading authority in functional analysis and operator semigroup theory. A doctoral student of H. H. Schaefer, he became internationally known for his work on one-parameter semigroups, positivity, and evolution equations. He served in major editorial roles in the field and initiated several international research and seminar programs in evolution equations and operator semigroups.
Frank Neubrander is an alumni professor of mathematics at Louisiana State University and an Executive Director and Chair of the Gordon A. Cain Center for STEM Literacy. He received his Ph.D. from the University of Tübingen in 1984 under Rainer Nagel and joined LSU in 1989. His research interests include evolution equations, Laplace transform methods, and mathematics education, and he has supervised numerous graduate students and led STEM education initiatives.
Content
Part I. One-parameter Semigroups on Banach Spaces.- Chapter 1. Basic results on Semigroups on Banach Spaces.- Chapter 2. Characterization of Semigroups on Banach Spaces.- Chapter 3. Spectral Theory.- Chapter 4. Asymptotics of Semigroups on Banach Spaces.- Part II. Positive Semigroups on Spaces C 0 ( X ) .- Chapter 5. Basic results on Spaces C 0 ( X ) .- Chapter 6. Characterization of Positive Semigroups on C 0 ( X ) .- Chapter 7. Spectral Theory of Positive Semigroups on C 0 ( X ) .- Chapter 8. Asymptotics of Positive Semigroups on C 0 ( X ) .- Part III. Positive Semigroups on Banach Lattices.- Chapter 9. Basic Results on Banach Lattices and Positive Operators.- Chapter 10. Characterization of Positive Semigroups on Banach Lattices.- Chapter 11. Spectral Theory on Banach Lattices.- Chapter 12. Asymptotics of Positive Semigroups on Banach Lattices.- Part IV. Positive Semigroups on C*- and W*- Algebras.- Chapter 13. Basic Results on Semigroups and Operator Algebras.- Chapter 14. Characterization of Positive Semigroups on W*-Algebras.- Chapter 15. Spectral theory of Positive Semigroups on W*-algebras and their Preduals.- Chapter 16. Asymptotics of Positive Semigroups on C*-and W*-Algebras.