On coefficients of vector-valued Bloch functions

Oscar Blasco. "On coefficients of vector-valued Bloch functions." Studia Mathematica 165.2 (2004): 101-110. <http://eudml.org/doc/284807>.

@article{OscarBlasco2004,
abstract = {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.},
author = {Oscar Blasco},
journal = {Studia Mathematica},
keywords = {Bloch functions; multiplier; Rademacher type and cotype; Fourier type; Orlicz property},
language = {eng},
number = {2},
pages = {101-110},
title = {On coefficients of vector-valued Bloch functions},
url = {http://eudml.org/doc/284807},
volume = {165},
year = {2004},
}

TY - JOUR
AU - Oscar Blasco
TI - On coefficients of vector-valued Bloch functions
JO - Studia Mathematica
PY - 2004
VL - 165
IS - 2
SP - 101
EP - 110
AB - 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.
LA - eng
KW - Bloch functions; multiplier; Rademacher type and cotype; Fourier type; Orlicz property
UR - http://eudml.org/doc/284807
ER -