Composition operators on Banach spaces of formal power series

Let {beta(n)}(n=infinity)(0) be a sequence of positive numbers and 1 less than or equal to p < infinity. We consider the space H-p(beta) of all, power series f(z) = Sigma(n=0)(infinity) (f) over cap (n) z(n) such that Sigma(n=0)(infinity) \(f) over capn\(p) beta(n)(p) < infinity. Suppose that 1/p + 1/q = 1 and Sigma(n=1)(infinity) n(qj)/beta(n)(q) = infinity for some non-negative negative integer j. We show that if C., is compact on H-P(beta), then the non-tangential limit of phi((j+1)) has modulus greater than one at each boundary point of the open it-nit disc. Also we show that if C-phi is Fredholm on H-P(beta), then phi must be an automorphism of the open unit disc.