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Multipliers on Vector Valued Bergman Spaces

Published online by Cambridge University Press:  20 November 2018

Oscar Blasco  and
José Luis Arregui
Oscar Blasco
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain, email: [email protected]
José Luis Arregui
Affiliation:
Departamento de Matemáticas, Universidad de Zaragoza, 50005 Zaragoza, Spain, email: [email protected]
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Abstract

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Let $X$ be a complex Banach space and let ${{B}_{p}}\left( X \right)$ denote the vector-valued Bergman space on the unit disc for $1\,\le \,p\,<\,\infty $. A sequence ${{\left( {{T}_{n}} \right)}_{n}}$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between ${{B}_{p}}\left( X \right)$ and ${{B}_{q}}\left( Y \right)$ (resp. ${{B}_{p}}\left( X \right)$ and ${{l}_{q}}\left( Y \right)$) if for any function $f\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{x}_{n}}{{z}^{n}}$ in ${{B}_{p}}\left( X \right)$ we have that $g\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{T}_{n}}\left( {{x}_{n}} \right){{z}^{n}}$ belongs to ${{B}_{q}}\left( Y \right)$ (resp. ${{\left( {{T}_{n}}\left( {{x}_{n}} \right) \right)}_{n}}\,\in \,{{\ell }_{q}}\left( Y \right)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in ${{B}_{p}}\left( X \right)$ are introduced.

Keywords

42A4546E40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 6 , 01 December 2002 , pp. 1165 - 1186
Copyright
Copyright © Canadian Mathematical Society 2002

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