Multipliers of the Hardy space H 1 and
نویسنده
چکیده
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 makes use of a new class of Fourier multipliers of H 1 which we call " shift-bounded ". We show that there is a ϕ which is a " completely bounded " multiplier of H 1 , or equivalently for which (ϕ(i + j)) i,j≥0 is a bounded Schur multiplier of B(ℓ 2), but which is not " shift-bounded " on H 1. We also give a characterization of " completely shift-bounded " multipliers on H 1 .
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