A primer on spectral theory
It is a compressed textbook on spectral theory of Banach algebras. After collecting some basic results in functional analysis and in the operator theory of Banach- and Hilbert-spaces, one basic part in this book (Chapter III) is the introduction into Banach algebras and the corresponding spectral theory. Many examples of Banach algebras are given, the spectrum and the spectral radius are defined. Analytic properties of the spectrum are discussed using the theory of subharmonic functions (spectrum maximum principle, Liouville's spectral theorem, subharmonicity of the n-th diameter, scarcity of elements with finite spectrum). Furthermore, several applications of spectral subharmonicity are discussed in a more theoretical framework. That covers for instance spectral characterizations of Banach algebras and of the radical. For these applications it is explained the representation theory of commutative and non-commutative Banach algebras, \(C^*\)-algebras are introduced and their spectral theorem is described. Finally, it is included an introduction for analytic multifunctions and a compendium of results on subharmonic functions, capacity, and functions of several complex variables. The reader should be familiar with several fundamentals in algebra, functional analysis, spectral theory in Banach spaces. Then he can find a set of interesting generalizations known for instance in the special case of the spectral theory in Hilbert spaces. Every chapter contains a series of exercises. Unfortunately, there is no list of symbols used in the book. Therefore it is sometimes difficult to find correct interpretations or the correct meaning.
scarcity of elements with finite spectrum
spectral characterizations of Banach algebras
representation theory of commutative and non- commutative Banach algebras
scientific article; zbMATH DE number 48817