Operator space valued Hankel matrices

الأصل · EN

If E is an operator space, the non-commutative vector valued Lᵖ spaces Sᵖ[E] have been defined by Pisier for any 1 ≤ p ≤ ∞. In this paper a necessary and sufficient condition for a Hankel matrix of the form (aᵢ₊ⱼ)₀ ≤ ᵢ,ⱼ with aₖ ∈ E to be bounded in Sᵖ[E] is established. This extends previous results of Peller where E= or E=Sᵖ. The main theorem states that if 1 ≤ p < ∞, (aᵢ₊ⱼ)₀ ≤ ᵢ,ⱼ is bounded in Sᵖ[E] if and only if there is an analytic function ϕ in the vector valued Besov Space Bₚ¹/ᵖ(E) such that aₙ = ϕ(n) for all n ∈. In particular this condition only depends on the Banach space structure of E. We also show that the norm of the isomorphism ϕ (ϕ(i+j))ᵢ,ⱼ grows as √ p as p → ∞, and compute the norm of the natural projection onto the space of Hankel matrices.