Banach Spaces and Operators

Abstract

The central topic of this book is the asymptotic distribution of the eigenvalues of certain compact (or more generally Riesz) operators in Banach spaces. We are interested in quantitative results: how fast do the eigenvalues of an operator T tend to zero if T belongs to some special class of compact operators? This is a classical subject in the context of integral operators where e.g. Hilbert-Schmidt kernels were studied. We replace these Hilbert-Schmidt operators by general ideals of compact or Riesz operators in Banach spaces, and describe some methods which were developed in the last years to prove general (upper) estimates for the eigenvalues of the operators belonging to such ideals. These abstract results are applied to integral operators to derive some non-classical results. The Banach space setting is essential: several applications, e.g. to Hille-Tamarkin kernels, have not been proved by the classical Hilbert space methods.