Modern Approaches to the Invariant-Subspace Problem

Presents work on the invariant subspace problem, a major unsolved problem in operator theory.

Isabelle Chalendar (Author), Jonathan R. Partington (Author)

9781107010512, Cambridge University Press

Hardback, published 18 August 2011

298 pages, 4 b/w illus. 65 exercises
22.9 x 15.2 x 2.1 cm, 0.61 kg

'I think this is a very useful book which will serve as a good source for a rich variety of methods that have been developed for proving positive results on the ISP. Moreover, there is much material in the book which is of interest beyond its application to the ISP. [It] should be of interest to analysts in general as well as being an essential source for study of the ISP.' Sandy Davie, SIAM Review

One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.

Introduction
1. Background
2. The operator-valued Poisson kernel and its applications
3. Properties (An,m) and factorization of integrable functions
4. Polynomially bounded operators with rich spectrum
5. Beurling algebras
6. Applications of a fixed-point theorem
7. Minimal vectors
8. Universal operators
9. Moment sequences and binomial sums
10. Positive and strictly-singular operators
Bibliography
Index.

Subject Areas: Calculus & mathematical analysis [PBK]