Multipliers of the Hardy space H 1 and power bounded operators Academic Article uri icon

abstract

  • © 2001, Instytut Matematyczny. All rights reserved. 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) = 〈Tnξ, η〉 This implies that the matrix (φ(i + j))i,j≥0is a Schur multiplier of B(ℓ2) or equivalently is in the space (ℓ1∨ ⊗ ℓ1) *. 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 H1, or equivalently for which (φ(i + j))i,j≥0is a bounded Schur multiplier of B(ℓ2), but which is not " shift-bounded " on H1. We also give a characterization of " completely shift-bounded " multipliers on H1

author list (cited authors)

  • Pisier, G.

citation count

  • 2

publication date

  • January 2001