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## Colloquium Mathematicum

2001 | 88 | 1 | 57-73
Tytuł artykułu

### Multipliers of the Hardy space H¹ and power bounded operators

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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.
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Tom
Numer
Strony
57-73
Opis fizyczny
Daty
wydano
2001
Twórcy
autor
• Texas A&M University, College Station, TX 77843, U.S.A.
• Equipe d'Analyse, Université Paris VI, Case 186, 75252 Paris Cedex 05, France
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