Charles Batty (Oxford) - "Quasi-hyperbolic operators and semigroups"

Abstract

An operator on a Banach space is hyperbolic (aka exponentially dichotomous) if it is a direct sum of two operators, the powers of one summand decays exponentially and the powers of the inverse of the other summand decay exponentially. Quasi-hyperbolic operators are not necessarily hyperbolic but they behave similarly. They arise naturally in pure operator theory, PDEs and differential geometry. The talk will start with the theory for single operators, which is quite compact. For one-parameter semigroups of operators the failure of spectral mapping theorems prevents a simple characterisation of quasi-hyperbolicity in terms of the generator. The talk will conclude with a discussion of some properties which can be deduced from the appropriate conditions on the generator.