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

التفاصيل البيبلوغرافية
العنوان: On characterizations of a some classes of Schauder frames in Banach spaces
المؤلفون: Karkri, Rafik, Kabbaj, Samir, Lafdal, Hamad Sidi
سنة النشر: 2023
المجموعة: Mathematics
مصطلحات موضوعية: Mathematics - Functional Analysis, 46B04, 46B10, 46B15, 46B25, 46B45
الوصف: 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}_{\pi}Y$ (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).
نوع الوثيقة: Working Paper
URL الوصول: http://arxiv.org/abs/2307.09174
رقم الأكسشن: edsarx.2307.09174
قاعدة البيانات: arXiv