Cyclic Vectors in Banach Spaces of Analytic Functions

Abstract

In these three lectures we consider Banach spaces of analytic functions on plane domains. If the space admits the operator of multiplication by z, then it is of interest to describe the cyclic vectors for this operator, that is, those functions in the space with the property that the polynomial multiples of the function are dense. A necessary condition is that the function have no zeros; in general it is difficult to give necessary and sufficient conditions.