Advanced Search

Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${\left(\sum Fₙ\right)}_{l_p}$, where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_p$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_p$. This gives an answer to a question of W. B. Johnson. We also prove that if X is...

Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-${\ell }_q$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-${\ell }_p$-tree estimate. Then T factors through a subspace of ${\left(\sum Fₙ\right)}_{{\ell }_r}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $({\ell }_p,{\ell }_q)$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic lower-${\ell }_q$-tree...

We give a corrected proof of Theorem 2.10 in our paper “Commutators on ${\left(\sum {\ell }_q\right)}_p$” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

Let T be a bounded linear operator on $X={\left(\sum {\ell }_q\right)}_p$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.