Multipliers of the Hardy space H1and power bounded operators

DOI10.4064/CM88-1-6zbMATH Open0983.42005arXivmath/0009074OpenAlexW2964312280MaRDI QIDQ2717705

Gilles Pisier

Publication date: 17 June 2001

Published in: Colloquium Mathematicum (Search for Journal in Brave)

Abstract: We study the space of functions $phicolonNNoCC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $xi,eta$ in $H$ such that ${\displaystyle }$phi(n) = < T^nxi,eta>.${\displaystyle }$ This implies that the matrix $(phi(i+j){)}_{i,jge0}$ is a Schur multiplier of $B(el{l}_2)$ or equivalently is in the space $$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded. We show that there is a $phi$ which is a ``completely bounded multiplier of $H^1$, or equivalently for which $(phi(i+j){)}_{i,jge0}$ is a bounded Schur multiplier of $B(el{l}_2)$, but which is not ``shift-bounded on $H^1$. We also give a characterization of ``completely shift-bounded multipliers on $H^1$.

Full work available at URL: https://arxiv.org/abs/math/0009074

zbMATH Keywords

Hardy spaceHilbert spaceHaagerup tensor producttensor productSchur multiplierFourier multiplierpower bounded operatoroperator algebra normshift-bounded

Mathematics Subject Classification ID

Multipliers for harmonic analysis in several variables (42B15) Groups and semigroups of linear operators (47D03)

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