اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIsomorphic Schauder decompositions in certain Banach spaces
143
0
0.0
(
0
)
تأليف
Vitalii Marchenko
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use $ell_{Psi}$-Hilbertian and $infty$-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition in a Hilbert space and introduce the class of spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and $M$-cotype of a Banach space and study the properties of unconditional Schauder decompositions in spaces possessing certain geometric structure.
قيم البحث
اقرأ أيضاً
Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach spaces. We
show that our results give Paley-Wiener theorem for frames for Hilbert spaces.
Famous Naimark-Han-Larson dilation theorem for frames in Hilbert spaces states that every frame for a separable Hilbert space $mathcal{H}$ is image of a Riesz basis under an orthogonal projection from a separable Hilbert space $mathcal{H}_1$ which co
ntains $mathcal{H}$ isometrically. In this paper, we derive dilation result for p-approximate Schauder frames for separable Banach spaces. Our result contains Naimark-Han-Larson dilation theorem as a particular case.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which c
an not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
Let $mathcal{H}$ be a complex separable Hilbert space. We prove that if ${f_{n}}_{n=1}^{infty}$ is a Schauder basis of the Hilbert space $mathcal{H}$, then the angles between any two vectors in this basis must have a positive lower bound. Furthermore
, we investigate that ${z^{sigma^{-1}(n)}}_{n=1}^{infty}$ can never be a Schauder basis of $L^{2}(mathbb{T}, u)$, where $mathbb{T}$ is the unit circle, $ u$ is a finite positive discrete measure, and $sigma: mathbb{Z} rightarrow mathbb{N}$ is an arbitrary surjective and injective map.
Assume that $mathcal{I}$ is an ideal on $mathbb{N}$, and $sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mat
hcal{I}textrm{-convergent}right}$. In the category case, we assume that $mathcal{I}$ has the Baire property and $sum_n x_n$ is not unconditionally convergent, and we deduce that $A(mathcal{I})$ is meager. We also study the smallness of $A(mathcal{I})$ in the measure case when the Haar probability measure $lambda$ on ${0,1}^{mathbb{N}}$ is considered. If $mathcal{I}$ is analytic or coanalytic, and $sum_n x_n$ is $mathcal{I}$-divergent, then $lambda(A(mathcal{I}))=0$ which extends the theorem of Dindov{s}, v{S}alat and Toma. Generalizing one of their examples, we show that, for every ideal $mathcal{I}$ on $mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $lambda(A(Fin))=0$ and $lambda(A(mathcal{I}))=1$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
