Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier. "Multipliers of the Hardy space H¹ and power bounded operators." Colloquium Mathematicae 88.1 (2001): 57-73. <http://eudml.org/doc/283555>.

@article{GillesPisier2001,
abstract = {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.},
author = {Gilles Pisier},
journal = {Colloquium Mathematicae},
keywords = {power bounded operator; Hilbert space; Schur multiplier; Fourier multiplier; tensor product; Haagerup tensor product; operator algebra norm; Hardy space; shift-bounded},
language = {eng},
number = {1},
pages = {57-73},
title = {Multipliers of the Hardy space H¹ and power bounded operators},
url = {http://eudml.org/doc/283555},
volume = {88},
year = {2001},
}

TY - JOUR
AU - Gilles Pisier
TI - Multipliers of the Hardy space H¹ and power bounded operators
JO - Colloquium Mathematicae
PY - 2001
VL - 88
IS - 1
SP - 57
EP - 73
AB - 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.
LA - eng
KW - power bounded operator; Hilbert space; Schur multiplier; Fourier multiplier; tensor product; Haagerup tensor product; operator algebra norm; Hardy space; shift-bounded
UR - http://eudml.org/doc/283555
ER -