On coefficients of vector valued Bloch functions

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc verifying that sup|z|<1(1−|z|)||f ′(z)|| < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and 1(Y ) if the map ∑∞ n=0 xnz n → (Tn(xn))n defines a bounded linear operator from Bloch(X) into 1(Y ). It is shown that if X is a Hilbert space then (Tn)n is a multiplier from Bloch(X) into 1(Y ) if and only supk ∑2k+1 n=2k ||Tn|| < ∞. Several results about Taylor coefficient of vector-valued Bloch functions depending on properties on X, such as Rademacher and Fourier type p, are presented. AMS Subj. Class: 46E40, 46B20