Theory of Semigroups and Applications
- 2017
- Buch
- 1. Auflage
- Verfasst von
- Kalyan B. Sinha
- Sachi Srivastava
- Buchreihe
- Texts and Readings in Mathematics
- Verlag
- Springer Singapore
Über dieses Buch
The book presents major topics in semigroups, such as operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics, and applications. Along with a systematic development of the subject, the book emphasises on the explorations of the contact areas and interfaces, supported by the presentations of explicit computations, wherever feasible. Designed into seven chapters and three appendixes, the book targets to the graduate and senior undergraduate students of mathematics, as well as researchers in the respective areas. The book envisages the pre-requisites of a good understanding of real analysis with elements of the theory of measures and integration, and a first course in functional analysis and in the theory of operators.
Chapters 4 through 6 contain advanced topics, which have many interesting applications such as the Feynman–Kac formula, the central limit theorem and the construction of Markov semigroups. Many examples have been given in each chapter, partly to initiate and motivate the theory developed and partly to underscore the applications. The choice of topics in this vastly developed book is a difficult one, and the authors have made an effort to stay closer to applications instead of bringing in too many abstract concepts.
Inhaltsverzeichnis
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Frontmatter
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Chapter 1. Vector-Valued Functions
Kalyan B. Sinha, Sachi SrivastavaAbstractThis chapter is mostly of a preliminary nature. In the first section we collect results on measurability and integrability of vector-valued functions that will be useful throughout. -
Chapter 2. -semigroups
Kalyan B. Sinha, Sachi SrivastavaAbstractIn this chapter we concentrate on strongly continuous or more specifically \(C_o\)-semigroups of bounded operators on a Banach space. The notion of the generator of a \(C_o\)-semigroup is introduced and their properties are dealt with in detail. -
Chapter 3. Dissipative Operators and Holomorphic Semigroups
Kalyan B. Sinha, Sachi SrivastavaAbstractIn this chapter we continue the study of \(C_0\)-semigroups concentrating on contractive and holomorphic semigroups. -
Chapter 4. Perturbation and Convergence of Semigroups
Kalyan B. Sinha, Sachi SrivastavaAbstractIn this chapter, the stability of various classes of semigroups under suitable sets of perturbations will be studied, viz. for general \(C_0\)-semigroups and contraction semigroups. -
Chapter 5. Chernoff’s Theorem and its Applications
Kalyan B. Sinha, Sachi SrivastavaAbstractIn this chapter, a very interesting theorem, due to Chernoff [4], is proven and some of its applications, viz. the Trotter-Kato Product Formula, the Feynman-Kac Formula and the Central Limit Theorem are given. -
Chapter 6. Markov Semigroups
Kalyan B. Sinha, Sachi SrivastavaAbstractIn Chapter 4, we have seen methods of constructing a semigroup by perturbing a known (or given) one, where the perturbation is small in a certain sense. More precisely, the context were those in which the candidate for the generator of the new semigroup was obtained by a small additive perturbation of the generator of the known semigroup. -
Chapter 7. Applications to Partial Differential Equations
Kalyan B. Sinha, Sachi SrivastavaAbstractWhile semigroups of operators are interesting objects of study by themselves, one of the main reasons why they are studied so extensively is due to the important role they play in the study of partial differential equations. -
Backmatter
- Titel
- Theory of Semigroups and Applications
- Verfasst von
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Kalyan B. Sinha
Sachi Srivastava
- Copyright-Jahr
- 2017
- Verlag
- Springer Singapore
- Electronic ISBN
- 978-981-10-4864-7
- DOI
- https://doi.org/10.1007/978-981-10-4864-7
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