Related papers: On characterizations of a some classes of Schauder…

Schauder bases are fundamental tools for analyzing the structure of Banach spaces. In this work, we show that Besselian Schauder frames (BSF) play a similar role in certain contexts. We first prove that every unconditional Schauder frame…

Let $H$ be an infinite-dimensional Hilbert space. We prove that every unconditional Schauder frame for $H$ contains a subsequence that can be normalized to form a frame for $H$. As a consequence, every semi-normalized unconditional Schauder…

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and…

We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames and coherent states for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. As…

For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains either $l_2$ or a subspace without…

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set $\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally saturated…

Let $X$ be a Banach space with an unconditional finite-dimensional Schauder decomposition $(E_n)$. We consider the general problem of characterizing conditions under which one can construct an unconditional basis for $X$ by forming an…

This paper brings new results on the FPP in Banach spaces $X$ with a Schauder basis. We first deal with the problem of whether there is a Banach space isomorphic to $\co$ having the FPP. We show that the answer is negative if $X$ contains a…

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and…

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\in\omega}$ for $X$, having the unconditional basic…

We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame…

In the present note we give a construction (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over a net $N$ in any separable infinite dimensional $\mathcal{L}_\infty$-space $X$. In…

In this paper, we introduce, for a separable Banach spacea new notion of besselian paires and of besselian Schauder frames for which we prove for some fundamental results.

We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable…

We prove that $L_2(\mathbb{R})$ contains a Schauder basis of non-negative functions. Similarly, $L_p(\mathbb{R})$ contains a Schauder basic sequence of non-negative functions such that $L_p(\mathbb{R})$ embeds into the closed span of the…

We present an isometric version of the complementably universal Banach space $\mathcal{B}$ with a monotone Schauder basis. The space $\mathcal{B}$ is isomorphic to Pe{\l}czy\'nski's space with a universal basis as well as to Kadec'…

Using methods of descriptive set theory, in particular, the determinacy of infinite games of perfect information, we answer several questions from the literature regarding different notions of bases in Banach spaces and lattices. For the…

Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame.…

In this paper we survey known results of characterizations of reflexive Banach spaces, which are based on convergence of usual and generalized arithmetic mean (or Ces\`aro sum), weakly compact subsets, affine sets in a Banach space or its…

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…