We study the space of functions ϕ: IN → C such that there is a Hilbert space H, a power bounded operator T in B(H) and vectors ξ, η in H such that ϕ(n) = T n ξ, η. This implies that the matrix (ϕ(i + j)) i,j≥0 is a Schur multiplier of B(ℓ 2) or equivalently is in the space (ℓ 1 ∨ ⊗ ℓ 1) *. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach make...