Multipliers of the Hardy space H¹ and power bounded operators

• Autorzy: Gilles Pisier
• Języki publikacji: EN

Abstrakty

EN

We study the space of functions φ: ℕ → 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 $(φ(i+j))_{i,j≥0}$ is 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 H¹ which we call "shift-bounded". We show that there is a φ which is a "completely bounded" multiplier of H¹, or equivalently for which $(φ(i+j))_{i,j≥0}$ is 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.

Kategorie tematyczne

• 42B15: Multipliers
• 47D03: Groups and semigroups of linear operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 88
• Numer: 1
• Strony: 57-73

Daty

wydano

2001

Twórcy

autor

Gilles Pisier

• Texas A&M University, College Station, TX 77843, U.S.A.
• Equipe d'Analyse, Université Paris VI, Case 186, 75252 Paris Cedex 05, France

Identyfikatory

DOI

10.4064/cm88-1-6