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

We study the space of functions $\phi\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $\xi,\eta$ in $H$ such that $$\phi(n) = < T^n\xi,\eta>.$$ This implies that the matrix $(\phi(i+j))_{i,j\ge 0}$ 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 $\phi$ which is a ``completely bounded'' multiplier of $H^1$, or equivalently for which $(\phi(i+j))_{i,j\ge 0}$ 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 $H^1$.