On coefficients of vector-valued Bloch functions
Volume 165 / 2004
Abstract
Let $X$ be a complex Banach space and let $\mathop{\rm Bloch}(X)$ denote the space of $X$-valued analytic functions on the unit disc such that $\sup_{|z|<1}(1-|z|^2)\|f'(z)\|<\infty$. A sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$ and $Y$ is said to be an operator-valued multiplier between $\mathop{\rm Bloch}(X)$ and $\ell_1(Y)$ if the map $\sum_{n=0}^\infty x_nz^n\to(T_n(x_n))_n$ defines a bounded linear operator from $\mathop{\rm Bloch}(X)$ into $\ell_1(Y)$. It is shown that if $X$ is a Hilbert space then $(T_n)_n$ is a multiplier from $\mathop{\rm Bloch}(X)$ into $\ell_1(Y)$ if and only if $\sup_{k} \sum_{n=2^k}^{2^{k+1}}\|T_n\|^2<\infty$. Several results about Taylor coefficients of vector-valued Bloch functions depending on properties on $X$, such as Rademacher and Fourier type $p$, are presented.
