A Characterization of Subspaces and Quotients of Reflexive Banach Spaces with Unconditional Bases

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. Then we prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite dimensional decomposition embeds into a reflexive Banach space with an unconditional basis

Embeddings and factorizations of Banach spaces

One problem, considered important in Banach space theory since at least the 1970’s, asks for intrinsic characterizations of subspaces of a Banach space with an unconditional basis. A more general question is to give necessary and sufficient conditions for operators from Lp (2 < p < 1) to factor through `p. In this dissertaion, solutions for the above problems are provided. More precisely, I prove that for a reflexive Banach space, being a subspace of a reflexive space with an unconditional basis or being a quotient of such a space, is equivalent to having the unconditional tree property. I also show that a bounded linear operator from Lp (2 < p < 1) factors through `p if and only it satisfies an upper-(C, p)-tree estimate. Results are then extended to operators from asymptotic lp spaces

A characterization of Schauder frames which are near-Schauder bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of $c_0$. In particular, a Schauder frame of a Banach space with no copy of $c_0$ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of $c_0$. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.Comment: 12 page

Commutators on (∑ℓq)ℓ1

Let T be a bounded linear operator on X=(∑ℓq)ℓ1 with 1≤. q\u3c. ∞. T is said to be X-strictly singular if the restriction of T on any subspace of X that is isomorphic to X is not an isomorphism. It is shown that the unique proper maximal ideal in L(X) is the set of all X-strictly singular operators. With some more efforts, we prove that T is a commutator in L(X) if and only if for all non-zero λ∈C, the operator T- λ. I is not X-strictly singular. © 2013 Elsevier Inc

Linear embedding and factorizations of operators on Banach spaces

Embedding theory is one of the most important topics in the geometry of Banach spaces and factorizations of operators are the natural extensions. In this paper, we will first systematically introduce the historical results on isomorphic theory and present some of the recent progress in this direction. Then we will discuss related results about factorizations of operators. Interesting open problems will be listed at the end of the paper

Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces

Non UBCUnreviewedAuthor affiliation: University of MemphisFacult

Operators on Lp (2 \u3c p \u3c ∞) which factor through Xp

Let T be a bounded linear operator on Lp (