On coefficients of vector-valued Bloch functions

Oscar Blasco

Studia Mathematica, Tome 162 (2004), p. 101-110 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that $sup_{|z|<1} (1 - |z|²)||f^{\prime }(z)|| < ∞$. 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 ℓ₁(Y) if the map $∑_{n=0}^{∞} xₙzⁿ → (Tₙ(xₙ))ₙ$ defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if $sup_{k} ∑_{n=2^{k}}^{2^{k+1}} ||Tₙ||² < ∞$. 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

Publié le : 2004-01-01

Zbl 1067.46039

EUDML-ID : urn:eudml:doc:284807