$L^p-$maximal regularity on Banach spaces with a Schauder basis

关于Banach空间中一类Schauder框架

On a class of Schauder frames in Banach spaces

本文研究Banach空间中besselian Schauder框架，给出besselian序列的刻画与性质，构造相应框架例子，并对具有此类框架的自反Banach空间给出若干刻画。

Isometric uniqueness of a complementably universal Banach space for Schauder decompositions

We present an isometric version of the complementably universal Banach space $\mathbb{P}$ with a Schauder decomposition. The space $\mathbb{P}$ is isomorphic to Pełczyński's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property.

Banach空间中几类Schauder框架的刻画

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

本文通过万有空间、有界逼近性质、张量积及Faber-Schauder系统等刻画Banach空间中的Schauder框架，并解决相关开放问题。

Expansion of approximate Bessel sequences to approximate Schauder frames for Banach spaces

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 can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.

Schauder bases in Lipschitz free spaces over nets in Banach spaces

In the present note we give two explicit constructions (based on a retractional argument) of a Schauder basis for the Lipschitz free space $\mathcal{F}(N)$, over certain uniformly discrete metric spaces $N$. The first one applies to every net $N$ in a finite dimensional Banach space, leading to the basis constant independent of the dimension. The second one applies to grids in Banach spaces with an FDD. As a corollary, we obtain a retractional Schauder basis for the Lipschitz free space $\mathcal{F}(N)$ over a net $N$ in every Banach space $X$ with a Schauder basis containing a copy of $c_0$, …

Characterising subspaces of Banach spaces with a Schauder basis having the shift property

We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht.

On general construct of chaotic unbounded linear operators in Banach Spaces with Schauder Bases

We utilize the idea underlying the construct of the classical weighted backward shift Rolewicz's operators to furnish a straightforward approach to a general construct of chaotic unbounded linear operators in a (real or complex) Banach space with a Schauder basis.

Upper and lower estimates for schauder frames and atomic decompositions

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 only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite dimensional Banach space has a Schauder frame, then it also has a Schauder frame which i…

On a characterization of convergence in Banach spaces with a Schauder basis

We extend the well-known characterizations of convergence in the spaces $l_p$ ($1\le p<\infty$) of $p$-summable sequence and $c_0$ of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space $c$ of convergent sequences.

一个具有绍德尔基的可分巴拿赫空间，它不是其二对偶空间的利普希茨收缩核

A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual

构造具有绍德尔基的可分巴拿赫空间\(X\)，证明\(B_X\)不是\(B_{X^{**}}\)的一致连续收缩核，得出\(X\)不是\(X^{**}\)的利普希茨收缩核的结论。

等距 universally 的 Banach 空间与单调 Schauder 基

An isometrically universal Banach space with a monotone Schauder basis

本文提出一个等距互补 universal Banach 空间 B，具有单调 Schauder 基，并与 Pełczyński 的 universal 基空间及 Kadec 的互补 universal 空间相关联。

$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

The main result of this paper is a fixed point result relating the spreading model structure of Banach spaces and Schauder basis with not too large basis constant. As a striking consequence, we deduce that every super-reflexive space has the fixed point property thus solving a long-standing open question in metric fixed point theory.

Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.

Approximately Dual p-Approximate Schauder Frames

Difficulty in the construction of dual frames for a given Hilbert space led to the introduction of approximately dual frames in Hilbert spaces by Christensen and Laugesen. It becomes even more difficult in Banach spaces to construct duals. For this purpose, we introduce approximately dual frames for a class of approximate Schauder frames for Banach spaces and develop basic theory. Approximate duals for this subclass is completely characterized and its perturbation is also studied.

Weaving Schauder frames

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 if and only if there is a uniform constant $C\geq 1$ such that every weaving is a $C$-approximate Schauder frame. We also study weaving Schauder bases, where it is necessary to introduce two notions of weaving. On one hand, we can ask if two Schauder bases are woven when considered as Schauder frames with their biorthogonal functionals, and alternatively, w…

Towards characterizations of approximate Schauder frame and its duals for Banach spaces

We begin the study of characterizations of recently defined approximate Schauder frame (ASF) and its duals for separable Banach spaces. We show that, under some conditions, both ASF and its dual frames can be characterized for Banach spaces. We also give an operator-theoretic characterization for similarity of ASFs. Our results encode the results of Holub, Li, Balan, Han, and Larson. We also address orthogonality of ASFs.

A new and simple proof of Schauder's theorem

Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).

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…

A Schauder basis for $L_2$ consisting of non-negative functions

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 sequence. We prove as well that if $X$ is a separable Banach space with the bounded approximation property, then any set in $X$ with dense span contains a quasi-basis (Schauder frame) for $X$. Furthermore, if $X$ is a separable Banach lattice with a bibasis then any set in $X$ with dense span contains a u-frame.

Dilation theorem for p-approximate Schauder frames for separable Banach 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 contains $\mathcal{H}$ 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.

希尔伯特空间与巴拿赫空间中的几乎Auerbach基、Markushevich基和Schauder基

Almost Auerbach, Markushevich and Schauder bases in Hilbert and Banach spaces

该研究构造了可分希尔伯特空间中满足特定级数发散条件的非强$(1+\varepsilon_n)$有界Markushevich基，证明条件的紧性，并推广到任意巴拿赫空间，得到经任意置换都不是Schauder基的对应基的存在性结论。

关于具有无条件Schauder基的Banach空间中FPP的注记

Remarks on the FPP in Banach spaces with unconditional Schauder basis

本文研究带Schauder基的Banach空间中的FPP与弱FPP，证明含等价于c₀单位基的预单调基本序列的空间无FPP，给出弱FPP的距离不变性及基相关充分条件。

Basis entropy in Banach spaces

We introduce and study two notions of entropy in a Banach space X with a normalized Schauder basis . The geometric entropy E(A) of a subset A of X is defined to be the infimum of radii of compact bricks containing A. We obtain several compactness characterizations for bricks (Theorem 3.7) useful for main results. We also obtain sufficient conditions on a set in a Hilbert space to have finite unconditional entropy. For Banach spaces without a Schauder basis we offer another entropy, called the Auerbach entropy. Finally, we pose some open problems.

Some remarks on smooth renormings of Banach spaces

We prove that in every separable Banach space $X$ with a Schauder basis and a $C^k$-smooth norm it is possible to approximate, uniformly on bounded sets, every equivalent norm with a $C^k$-smooth one in a way that the approximation is improving as fast as we wish on the elements depending only on the tail of the Schauder basis. Our result solves a problem from the recent monograph of Guirao, Montesinos and Zizler.

Lipschitz p-Approximate Schauder Frames

With the aim of representing subsets of Banach spaces as an infinite series using Lipschitz functions, we study a variant of metric frames which we call Lipschitz p-approximate Schauder frames (Lipschitz p-ASFs). We characterize Lipschitz p-ASFs and their duals completely using the canonical Schauder basis for classical sequence spaces. Similarity of Lipschitz p-ASF is introduced and characterized.

New Identity on Parseval p-Approximate Schauder Frames and Applications

A very useful identity for Parseval frames for Hilbert spaces was obtained by Balan, Casazza, Edidin, and Kutyniok. In this paper, we obtain a similar identity for Parseval p-approximate Schauder frames for Banach spaces which admits a homogeneous semi-inner product in the sense of Lumer-Giles.

Amalgamations of classes of Banach spaces with a monotone basis

It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a…

Weak-star point of continuity property and Schauder bases

We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in \cite{DF} obtained for the separable case. Also, as a consequence o…

Subsymmetric weak$^*$ Schauder bases and factorization of the identity

Let $X^*$ denote a Banach space with a subsymmetric weak$^*$ Schauder basis satisfying condition~\eqref{eq:condition-c}. We show that for any operator $T : X^*\to X^*$, either $T(X^*)$ or $(I-T)(X^*)$ contains a subspace that is isomorphic to $X^*$ and complemented in $X^*$. Moreover, we prove that $\ell^p(X^*)$, $1\leq p \leq \infty$ is primary.

Schauder-Orlicz 分解、$\ell_Φ$-分解与伪 Daugavet 性质

Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property

本文推广了Banach空间的$\ell_p$-分解概念，提出$\ell_Φ$-分解与Schauder-Orlicz分解，并证明在具有伪Daugavet性质的空间中不存在包含有限维子空间的此类分解。

Feichtinger Conjectures, $R_\varepsilon$-Conjectures and Weaver's Conjectures for Banach spaces

Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality f…

关于Banach空间条件基对称类型的注记

Remarks about symmetry-type conditions of conditional bases of Banach spaces

该研究探讨了准Banach空间中条件基的对称性和扩散性，通过重范数构造使排列和递增映射成为等距映射，扩展了经典等距重范数定理的应用范围。

Schauder basis in Lipschitz free spaces over nets of $\mathcal{L}_\infty$-spaces

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 particular, this yields the first example of an infinite dimensional Banach space $X$ not containing $c_0$ with such a property.

The $(abc,pqr)$-problem for Approximate Schauder Frames for Banach Spaces

Motivated from the complete solution of important $abc$-problem for Gabor system for the Hilbert space $\mathcal{L}^2(\mathbb{R})$ by Dai and Sun [\textit{Memoirs of Amer. Math. Soc., 2016}] and from the existential result of approximate Schauder frames for $\mathcal{L}^p(\mathbb{R})$ using translation operators on $\mathcal{L}^p(\mathbb{R})$ by Freeman, Odell, Schlumprecht, and Zsak [\textit{Israel. J. Math, 2014}], we formulate $(abc, pqr)$-problem for approximate Schauder frames for Banach spaces $\mathcal{L}^p(\mathbb{R})$, $1<p<\infty$.

Banach空间和格中的坐标系统

Coordinate systems in Banach spaces and lattices

本文利用描述集合论方法，回答了关于Banach空间和格中不同基概念的问题。证明了在Banach格中，σ-序基在分析确定性下是均匀基，且均匀基是Schauder基。同时，当X=[e_n]时，序基与σ-序基概念一致。对于Banach空间，研究了滤子Schauder基的相关问题。

Hereditarily Indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces $X$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to $\mathcal{L}_{diag}({\mathfrak X}_D)$ with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every $T\in \mathcal{L}_{dia…

Compact retractions and Schauder decompositions in Banach spaces

In our note we show the very close connection between the existence of a Finite Dimensional Decomposition (FDD for short) for a separable Banach space $X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a certain precise sense) generating convex and compact subset $K$ of $X$. In one direction, if $X$ admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set $K$. On the other hand, we prove that if $X$ admits a small generating compact Lipschitz retract then $X$ has the $π$-property. We note that it is still unknown if the $π$-prop…

Shrinking Schauder Frames and their Associated Bases

For a Banach space $X$ with a shrinking Schauder frame $(x_i,f_i)$ we provide an explicit method for constructing a shrinking associated basis. In the case that the minimal associated basis is not shrinking, we prove that every shrinking associated basis of $(x_i,f_i)$ dominates an uncountable family of incomparable shrinking associated bases of $(x_i,f_i)$. By adapting a construction of Pełczy{ń}ski, we characterize spaces with shrinking Schauder frames as spaces having the $w^*$-bounded approximation property.

A universal Banach space with a $K$-unconditional basis

For a constant $K\geq 1$ let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\inω})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\inω}$ for $X$, having the unconditional basic constant $K_u\leq K$. Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$. Using the technique of Fraïssé theory, we construct a rational $K$-based Banach space $…

Hilbert-Schauder Frame Operators

We introduce a new concept of frame operators for Banach spaces we call a Hilbert-Schauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we give a new isomorphic characterization of Hilbert spaces.

Banach spaces without local unconditional structure

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 unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow.

可分希尔伯特空间中的Schauder正则性结果

Schauder regularity results in separable Hilbert spaces

本文在可分希尔伯特空间中，针对与半线性随机偏微分方程相关的转移半群弱生成元驱动的平稳与演化方程，证明其解的Schauder型正则性估计。

The universal Banach space with a $K$-suppression unconditional basis

Using the technique of Fraïssé theory, for every constant $K\ge 1$ we consruct a universal object in the class of Banach spaces with normalized $K$-suppression unconditional Schauder bases.

论 Auerbach 或几乎 Auerbach Markushevich 系与 Schauder 基之间的关系

On the relations between Auerbach or almost Auberbach Markushevich systems and Schauder bases

本文研究可分Hilbert空间中几乎Auerbach Markushevich基与Riesz基、Schauder基的关系，证明级数可和性是充要条件，并构造了任意置换下不等价于Schauder基的Auerbach基例子。

自反 Banach 空间的刻画

Characterizations of reflexive Banach spaces

本文综述自反 Banach 空间基于收敛性、弱紧性、仿射集和双正交系统的刻画，并讨论含 Schauder 基空间中与 $\ell^1$、$c_0$ 同构及非自反性的关系。

理想 Schauder 基的动物园

Zoo of ideal Schauder basis

本文研究 Banach 空间中的滤子（等价于理想）Schauder 基，通过在经典空间中构造大量非标准新例子，刻画不同滤子/理想并分析其与基本序列的关系。

Relative compactness of orbits and geometry of Banach spaces

We investigate for a bounded semigroup of linear operators $S$ on a Banach space $E$ and a vector $x \in E$, when relative compactness of $S(I-T)x$ for every $T \in S$ implies relative compactness of the orbit $Sx$. In particular, we derive characterizations of separable Banach spaces not containing $\mathrm{c}_0$ and of reflexivity of Banach spaces with a Schauder basis in terms of such compactness results.

Non-universal families of separable Banach spaces

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 Banach space with a monotone Schauder basis which is isometrically universal for $ C $ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.

度量框架、Schauder框架与算子值框架

Metric, Schauder and Operator-Valued Frames

本文系统引入并研究度量框架、近似Schauder框架及广义算子值框架，给出其刻画、对偶、稳定性、相似性与插值等结果，并建立与Lipschitz算子和群酉表示的联系。

具有许多好块基本序列的 Schauder 基

Schauder Bases Having Many Good Block Basic Sequences

本文研究 Schauder 基中好块基本序列的丰富性，刻画块树分支皆好与空间为 $1$-渐近 $\ell_p$ 的等价性，并借助 Hindman-Milliken-Taylor 定理证明常系数块基本序列的稳定化结果。

Monomial basis in Korenblum type spaces of analytic functions

It is shown that the monomials $Λ=(z^n)_{n=0}^{\infty}$ are a Schauder basis of the Fréchet spaces $A_+^{-γ}, \ γ\geq 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|)^μ|f(z)|$ is bounded for all $μ> γ$. Lusky \cite{L} proved that $Λ$ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $H^{\infty}$. A sequence space representation of the Fréchet space $A_+^{-γ}$ is presented. The case of (LB)-spaces $A_{-}^{-γ}, \ γ> 0,$ that are defined as unions of weighted Banach spaces is also studied.

Arens regularity of module actions and weak amenability of Banach algebras

For Banach left and right module actions, we will establish the relationships between topological centers of module actions with some result in the weak amenability of Banach algebras.

A Banach space dichotomy for quotients of subspaces

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable (QHI) property if $X/Y$ is hereditarily indecomposable (HI) for any infinite codimensional subspace $Y$ with a successive finite-dimensional decomposition on the basis of $X$. A reflexive space with the restricted QHI property is in particular HI, has HI dual, and is saturated with subspaces which are HI and have HI dual. The following dichotomy theorem is proved: any infinite dimensional Banach space contains a quotient of subspace which either has an unconditional basis, or has the…

具有无条件基的巴拿赫空间上全纯函数的单项式基

Monomial bases for holomorphic functions on Banach spaces with an unconditional basis

本文针对具有无条件基的复巴拿赫空间，证明特定排序的单项式构成其带紧开拓扑的整全纯函数空间的Schauder基，并将结果应用于多类巴拿赫序列格与向量值E-和空间。