Multipliers of the Hardy space $H^1$ and power bounded operators
Volume 88 / 2001
Abstract
We study the space of functions $\varphi :{\mathbb N}\to {\mathbb C}$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $\xi ,\eta $ in $H$ such that $\varphi (n) = \langle T^n\xi ,\eta \rangle .$ 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 $(\ell _1 \mathrel {\breve {\otimes }} \ell _1)^*$. 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 $H^1$.
