Multipliers of the Hardy space H^1 and power bounded operators

Abstract

We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $\xi ,\eta$ in $H$ such that $$\varphi (n)=<T^n\xi ,\eta >.$$ This implies that the matrix $(\varphi (i+j){)}_{i,j\ge 0}$ is a Schur multiplier of $B({\ell }_2)$ or equivalently is in the space $(\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*$. 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 $\varphi$ which is a ``completely bounded'' multiplier of $H^1$, or equivalently for which $(\varphi (i+j){)}_{i,j\ge 0}$ is a bounded Schur multiplier of $B({\ell }_2)$, but which is not ``shift-bounded'' on $H^1$. We also give a characterization of ``completely shift-bounded'' multipliers on $H^1$.

Submitted to Colloquium Math