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Rafik Karkri

Publications and source records attributed to Rafik Karkri.

6 recordsLinked to original sources
arXiv2025

A New Crossnorm That Preserves Unconditional Bases in Banach Spaces

Let $α$ be a tensor norm (i.e., a uniform reasonable crossnorm) on the class of all algebraic tensor products of Banach spaces $E \otimes F$. We say that $α$ preserves unconditionality if, for every pair of Banach spaces $E$ and $F$ with unconditional Schauder bases (USBs), the completion $E \otimes_α F$ also admits a USB. It is well known that none of Grothendieck's fourteen natural tensor norms satisfy this unconditionality-preserving condition. Moreover, the existence of a tensor norm $α$ with this property remains an open question. In this paper, we construct for every such pair $(E,F)$ a new reasonable crossnorm $α$. This norm has the surprising property that -- despite being generally non-uniform -- the space $E \otimes_α F$ nevertheless admits a USB.

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arXiv2024

Besselian Schauder Frames and the Structure of Banach Spaces

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. BSF are a new class of Schauder frames, lying between unconditional and general frames. We first prove that every unconditional Schauder frame (USF) is BSF, but the reverse implication is false. Specifically, we extend several well-known results of Karlin and James to Banach spaces with BSF, particularly to those with USF. We prove that many classical Banach spaces do not admit BSF, and in particular, do not admit USF. Before establishing these results, for every Banach space $E$ with a finite dimensional decomposition, we provide an explicit method to construct a Schauder frame for $E$. In particular, Szarek's Banach space has a Schauder frame, which famously lacks a Schauder basis. This finding provides strong motivation for extending classical Schauder basis theory to the framework of Schauder frames.

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arXiv2023

On characterizations of a some classes of Schauder frames in Banach spaces

In this paper, we prove the following results. There exists a Banach space without basis which has a Schauder frame. There exists an universal Banach space $B$ (resp. $\tilde{B}$) with a basis (resp. an unconditional basis) such that, a Banach $X$ has a Schauder frame (resp. an unconditional Schauder frame ) if and only if $X$ is isomorphic to a complemented subspace of $B$ (resp. $\tilde{B}$). For a weakly sequentially complete Banach space, a Schauder frame is unconditional if and only if it is besselian. A separable Banach space $X$ has a Schauder frame if and only if it has the bounded approximation property. Consequenty, The Banach space $\mathcal{L}(\mathcal{H},\mathcal{H})$ of all bounded linear operators on a Hilbert space $\mathcal{H}$ has no Schauder frame. Also, if $X$ and $Y$ are Banach spaces with Schauder frames then, the Banach space $ X\widehat{\otimes}_πY$ (the projective tensor product of $X$ and $Y$) has a Schauder frame. From the Faber$-$Schauder system we construct a Schauder frame for the Banach space $C[0,1]$ (the Banach space of continuous functions on the closed interval $ [0,1]$) which is not a Schauder basis of $C[0,1]$. Finally, we give a positive answer to some open problems related to the Schauder bases (In the Schauder frames setting).

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arXiv2023

On a class of Schauder frames in Banach spaces

In this paper, we give a characterization and a some properties of a besselian sequences, which allows us to build some examples of a besselian Schauder frames. Also for a reflexive Banach spaces (with a besselian Schauder frames) we give some characterizations.

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