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Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier — 2001

Colloquium Mathematicae

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...

Real Interpolation between Row and Column Spaces

Gilles Pisier — 2011

Bulletin of the Polish Academy of Sciences. Mathematics

We give an equivalent expression for the K-functional associated to the pair of operator spaces (R,C) formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair (Mₙ(R),Mₙ(C)) (uniformly over n). More generally, the same result is valid when Mₙ (or B(ℓ₂)) is replaced by any semi-finite von Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust-Piquard) that is valid for the Lorentz spaces...

Interpolation between H spaces and non-commutative generalizations (II).

Gilles Pisier — 1993

Revista Matemática Iberoamericana

We continue an investigation started in a preceding paper. We discuss tha classical result of Carleson connecting Carleson measures with the ∂-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution for the ∂-equation, which satisfies simultaneously a good L estimate and a good L1 estimate. This appears as a special case of our main result which can be stated as...

Une nouvelle classe d'espaces de Banach vérifiant le théorème de Grothendieck

Gilles Pisier — 1978

Annales de l'institut Fourier

Soit W un espace 1 et soit R un sous-espace réflexif de dimension infinie de W . Nous montrons que le quotient W / R vérifie le théorème de Grothendieck, c’est-à-dire que tout opérateur de W / R dans un espace de Hilbert est 1-sommant; par ailleurs, W / R n’est pas un espace 1 . Cela permet de répondre négativement à une question de Lindenstrauss-Pełczyński ainsi qu’à une question similaire de Grothendieck.

Quantum expanders and geometry of operator spaces

Gilles Pisier — 2014

Journal of the European Mathematical Society

We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of M N -spaces needed to represent (up to a constant C > 1 ) the M N -version of the n -dimensional operator Hilbert space O H n as a direct sum of copies of M N . We show that, when C is close to 1, this multiplicity grows as exp β n N 2 for...

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