On coefficients of vector-valued Bloch functions

Oscar Blasco ¹

¹ Departamento de Análisis Matemático Universidad de Valencia 46100 Burjassot Valencia, Spain

Studia Mathematica, Tome 165 (2004) no. 2, pp. 101-110

Résumé

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.

Détail

DOI : 10.4064/sm165-2-1

Keywords: complex banach space mathop bloch denote space x valued analytic functions unit disc sup infty sequence bounded operators between banach spaces said operator valued multiplier between mathop bloch ell map sum infty n defines bounded linear operator mathop bloch ell shown hilbert space multiplier mathop bloch ell only sup sum infty several results about taylor coefficients vector valued bloch functions depending properties rademacher fourier type presented