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.
Isabelle Chalendar is an Assistant Professor in the Department of Mathematics at the University of Lyon 1, France. Jonathan R. Partington is a Professor in the School of Mathematics at the University of Leeds.
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.