BELLWETHERS FOR BOUNDEDNESS OF COMPOSITION OPERATORS ON WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

Let D be the open unit disc, let nu : D -> (0, infinity) be a typical weight, and let H(nu)(infinity) be the corresponding weighted Banach space consisting of analytic functions f on D such that parallel to f parallel to(nu):=sup(z is an element of D) nu(z)vertical bar f (z)vertical bar < infinity. We call H(nu)(infinity) a typical-growth space. For phi a holomorphic self-map of D, let C(phi) denote the composition operator induced by phi. We say that C(phi) is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight nu, C(phi) acts boundedly on H(nu)(infinity) only if all composition operators act boundedly on H(nu)(infinity). We show that a sufficient condition for C(phi) to be a bellwether for boundedness is that phi have an angular derivative of modulus less than I at a point on partial derivative D. We raise the question of whether this angular-derivative condition is also necessary for C(phi) to be a bellwether for boundedness.