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 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-6,
author = {Gilles Pisier},
title = {Multipliers of the Hardy space H$^1$ and power bounded operators},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {57-73},
zbl = {0983.42005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-6}
}
Gilles Pisier. Multipliers of the Hardy space H¹ and power bounded operators. Colloquium Mathematicae, Tome 89 (2001) pp. 57-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-1-6/