63 research outputs found

### Perturbations of frames

In this paper, we give some sufficient conditions under which perturbations
preserve Hilbert frames and near-Riesz bases. Similar results are also extended
to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning
that some of our perturbation conditions are quite different from those used in
the previous literatures on this topic.Comment: to appear in Acta MAth. Sinica, English Serie

### 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 1970â€™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 (âˆ‘â„“q)â„“1

Let T be a bounded linear operator on X=(âˆ‘â„“q)â„“1 with 1â‰¤. q\u3c. âˆž. 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 Î»âˆˆC, the operator T- Î». I is not X-strictly singular. Â© 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 âˆž) which factor through Xp

Let T be a bounded linear operator on Lp (

### On operators from separable reflexive spaces with asymptotic structure

Let 1 \u3c q \u3c p \u3c âˆž and q â‰¤ r â‰¤ p. Let X be a reflexive Banach space satisfying a lower-â„“q-tree estimate and let T be a bounded linear operator from X which satisfies an upper-â„“p-tree estimate. Then T factors through a subspace of (Î£ Fn) â„“r, where (Fn) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an (â„“p, â„“q) FDD. Similarly, let 1 \u3c q \u3c r \u3c p \u3c âˆž and let X be a separable reflexive Banach space satisfying an asymptotic lower-â„“q-tree estimate. Let T be a bounded linear operator from X which satisfies an asymptotic upper-â„“p-tree estimate. Then T factors through a subspace of (Î£ Gn) â„“r, where (Gn) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an asymptotic (â„“p, â„“q) FDD. Â© Instytut Matematyczny PAN, 2008

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