Local Entropy Moduli and Eigenvalues of Operators in Banach Spaces.

In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators between lp-spaces can be estimated. It is shown, that in general local and global degree of compactness considerably differ, but under certain type assumptions on the underlying Banach spaces they coincide. Finally, the results are applied to obtain (optimal) estimates for eigenvalues of certain integral operators