Evolution semigroups and time operators on Banach spaces

We present new general methods to obtain shift representation of evolution semigroups defined on Banach spaces We introduce the notion of time operator associated with a generalized shift on a Banach space and give some conditions under which time operators can be defined on an arbitrary Banach space. We also tackle the problem of scaling of time operators and obtain a general result about the existence of time operators on Banach spaces satisfying some geometric conditions. The last part of the paper contains some examples of explicit constructions of time operators on function spaces. (C) 2010 Elsevier Inc All rights reserved.