Compositions of projections in Banach spaces and relations between approximation properties
M. I. Ostrovskii
给出了巴拿赫空间存在有限维分解但不具有投影合成范数的$π$ -性质的一个充要条件.
A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $π$-property in terms of norms of compositions of projections is found.
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In general Banach spaces, the metric projection map lacks the powerful properties it enjoys in Hilbert spaces. There are a few generalized projections that have been proposed in order to resolve many of the deficiencies of the metric projection. However, such notions are predominantly studied in Banach spaces with rich topological structures, such as uniformly convex Banach spaces. In this paper, we investigate two notions of generalized projection in general Banach spaces. Various examples are provided to demonstrate the proposed notions and the loss of structure in the generalized projection…
Representation of Generalized Bi-Circular Projections on Banach Spaces
本文证明了任意巴拿赫空间上投影表示的几个结果。我们还给出了一些说明性的例子，包括一个广义双圆投影的例子，它不能写成等距反射恒等式的平均。我们还刻画了$C_0(\Om,X)$上的广义双圆投影，其中$\Om$是局部紧T2空间（不一定连通），$X$是具有平凡中心化子的巴拿赫空间.
A Strong Convergence Theorem for finite families of Bregman Demimetric Mappings in a Banach Space under a New Shrinking projection Method
We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.
Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections
本文利用度量投影证明了一致凸光滑巴拿赫空间中渐近非扩张映象的一个强收敛定理。这个定理推广和改进了Matsushita和Takahashi [Appl. Math. Comput. 196（2008）422-425]，其是针对非扩张映射建立的。
Spectrum of Weighted Composition Operators. Part II. Weighted composition operators on subspaces of Banach lattices
我们描述了频谱的加权$d$ -同构的巴拿赫格限制在封闭的子空间是“丰富”的，足以保持一些“记忆”的顺序结构的原始格。例子包括（但不限于）多圆盘上的Hardy空间的加权等距和$\mathds{C}^n$中的单位球。
We study composition operators on the weighted Banach spaces of an infinite tree. We characterize the bounded and the compact operators, as well as determine the operator norm and the essential norm. In addition, we study the isometric composition operators.
Norm One Projections in Banach Spaces
本文综述了Köthe函数空间和Banach数列空间中关于范数1投影和1-补子空间的一些结果。从20世纪30年代到最新思想的发展，介绍了这一理论的历史发展。主要结果的证明概述。开放的问题也进行了讨论。已尽一切努力列入尽可能完整的参考书目。
Band projections and order idempotents in Banach lattice algebras
本文在Banach格代数中引入带投影概念，研究其与序幂等元之间的关系，并重新考察由单位元生成的理想的性质。
Mordukhovich Derivatives and Covering Constant for Set-Valued Metric Projection in General Banach Spaces
本文得到l₁空间到其单位闭球的集值度量投影算子的显式表示，研究其Mordukhovich导数性质以计算覆盖常数，还针对二维巴拿赫空间得到该投影算子的显式解并详细计算其导数。
In this paper, we study the hypercyclic composition operators on weighted Banach spaces of functions defined on discrete metric spaces. We show that the only such composition operators act on the "little" spaces. We characterize the bounded composition operators on the little spaces, as well as provide various necessary conditions for hypercyclicity.
Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blo…
Shrinking projection method for a sequence of relatively quasi-nonexpansive multivalued mappings and equilibrium problem in Banach spaces
在巴拿赫空间中建立了一个新的基于收缩投影方法的迭代程序对相对拟非扩张集值映象无限族的公共不动点的公共元和平衡问题的解集的强收敛性.我们的结果改进和推广了许多人的相应结果。
Frechet differentiability of the metric projection operator in Banach spaces
本文研究了一致凸且一致光滑的 Banach 空间中度量投影算子的 Frechet 可微性，并针对闭球、闭凸圆柱和正锥等闭凸子集，求得了其 Frechet 导数与 Gateaux 方向导数的精确表达式。
Mordukhovich derivatives of the metric projection operator in uniformly convex and uniformly smooth Banach spaces
该研究考察一致凸且一致光滑 Banach 空间中度量投影算子的 Mordukhovich 导数，并给出其在闭球、闭凸柱体和正锥情形下的精确表达式。
We present the current results in the study of weighted composition operators on weighted Banach spaces of an unbounded, locally finite metric space. Specifically, we determine characterizations of bounded and compact weighted composition operators, including the operator and essential norms. In addition, we characterize the weighted composition operators that are injective, are bounded below, have closed range, and have bounded inverse. We characterize the isometries and surjective isometries among the weighted composition operators, as well as those that satisfy the Fredholm condition. Lastl…
We study mean ergodic composition operators on infinite dimensional spaces of holomorphic functions of different types when defined on the unit ball of a Banach or a Hilbert space: that of all holomorphic functions, that of holomorphic functions of bounded type and that of bounded holomorphic functions. Several examples in the different settings are given.
The Amalgamation Property in Classical Lebesgue-Riesz Spaces, Banach Spaces with Transitive and Almost Transitive Norms and Projection Constants
本文证明了Lebesgue-Riesz空间Lp所生成的有限等价类具有合并性质当且仅当p不等于强大于2的偶数自然数.
Mordukhovich derivatives of the set-valued metric projection operator in general Banach spaces
本文研究一般 Banach 空间中集值度量投影算子的 Mordukhovich 导数性质与精确解，并证明了 C[0, 1] 到多项式空间的投影为单值映射，进而探讨了其 Mordukhovich 导数与 Gateaux 方向导数。
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide computable estimates on the operator norm.
Generalized 3-circular projections in some Banach spaces
最近在一系列的论文中发现，在许多巴拿赫空间中，包括经典空间$C(Ω)$和$L_p$ -空间$1 \leq p < \infty, p \neq 2$，任何广义双圆投影$P$由$P = \frac{I+T}{2}$给出，其中$I$是空间的恒等算子，$T$是一个反射，即$T$是与$T^2 = I$的满射等距.对于$n \geq 3$阶的满射等距，对应的投影概念是广义$n$ -圆投影，定义在\cite{AD}中。本文证明了在Banach空间$X$中，如果广义双圆投影由$\frac{I+T}{2}$给出，其中$T$是一个反射，那么任何广义$n$ -圆投影$P$，$n \geq 3$，由$P = \frac{I+T+T^2+...+T^{n-1}}{n}$给出，其中$T$是一个满射等距，而$T^n = I$ .我们证明了我们的$n=3$和$n > 3$的结果，证明保持不变，除了例行修改。
A New Shrinking projection Algorithm for an infinite family of Bregman weak relatively nonexpansive mappings in a Banach Space
提出一种结合广义预解式与广义投影的新型收缩投影算法，证明其在自反Banach空间中可强收敛到不动点、零点及平衡问题解的公共点。
Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces
根据von Neumann-Halperin和Lapidus定理，在希尔伯特空间中，正交投影的乘积或凸组合的迭代是强收敛的。我们将这些结果推广到复巴拿赫空间中某些投影乘积的凸组合的迭代。后者被假定为一致凸或一致光滑的正交投影，或自反的更特殊的投影，特别是，埃尔米特的。在所有情况下的收敛性证明是基于一个已知的标准的边界谱。
Observations on the metric projection in finite dimensional Banach spaces
本文研究有限维Banach空间中交替度量投影方法的收敛性，刻画了使该方法对任意初始点都收敛到交集投影的子空间对，并证明在 $\ell_p^3$ 中此类子空间对在概率意义上是稀少的。
Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let Pc from X to C denote the (standard) metric projection operator. In this paper, we define the Gateaux directional differentiability of Pc. We investigate some properties of the Gateaux directional differentiability of Pc. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of Pc. Finally, we define the concept of p-q uniformly convex and uniformly smooth Ban…
We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators among the bounded operators on such spaces, thus generalizing some well-known results on multiplication or composition operators. We also characterize the invertible weighted composition operato…
Directional Differentiability of the Generalized Metric Projection in Banach Spaces
本文研究实一致凸且一致光滑Banach空间中广义度量投影算子的Gateaux方向可微性，并针对闭球、闭凸锥和闭凸柱体给出方向导数的精确表示及与标准度量投影的比较。
Generalized Projection Operators in Banach Spaces: Properties and Applications
本文是一个帐户（没有证明）的结果，我们的工作“度量和广义投影算子在Banach空间：性质和应用”，funct-an/9311001。第九节利用度量算子和广义投影算子建立了Banach空间中变分不等式与Wienner-Hopf方程之间的联系。
Banach spaces with projectional skeletons
巴拿赫空间中的投影骨架是到可分离子空间上的σ-定向投影族，覆盖整个空间。具有投影骨架的巴拿赫空间类严格大于Plichko空间类（即具有可数赋范Markushevich基的巴拿赫空间）。我们证明了每一个具有投影骨架的空间都有一个投影分解的恒等式，并且有一个赋范空间，它具有与Sigma-空间相似的性质。我们刻画存在的一个基本的子结构的投影骨架，提供简单的证明已知的结果弱紧生成空间和Plichko空间。我们证明了Plichko巴拿赫空间的一个守恒结果，涉及超限序列的投影。作为推论，我们证明了一个巴拿赫空间是Plichko的当且仅当它有一个交换的投影骨架。
The general problem we address is to develop new methods in the study of projection constants of Banach spaces of multivariate polynomials. The relative projection constant $\boldsymbolλ(X,Y)$ of a subspace $X$ of a Banach $Y$ is the smallest norm among all possible projections on $Y$ onto $X$, and the projection constant $\boldsymbolλ(X)$ is the supremum of all relative projection constants of $X$ taken with respect to all possible super spaces $Y$. This is one of the most significant notions of modern Banach space theory and has been intensively studied since the birth of abstract operator t…
Global Version of Bjornestal's Estimate for Metric Projection Operator in Banach Space
1979年，B.Bjornestal在一致凸一致光滑Banach空间$B$中得到了度量投影算子在闭子空间上的一致连续模的局部估计.本文对$B$中任意闭凸集上的投影算子给出了这个结果的整体形式。
We show a way of constructing projectional skeletons using the concept of rich families in Banach spaces which admit a projectional generator. Our next result is that a Banach space $X$ is Asplund and weakly compactly generated if and only if there exists a commutative 1-projectional skeleton $(Q_γ:\ γ\inΓ)$ on $X$ such that $(Q_γ^*:\ γ\inΓ)$ is a commutative 1-projectional skeleton on $X^*$. We consider both, real and also complex, Banach spaces.
On Some Estimates for Projection Operator in Banach Space
得到了Banach空间中投影算子一致连续性的未知估计。它们可用于近似方法的研究，特别是拟解方法、正则化方法和罚函数方法，用于解决精确集和扰动集上的非线性问题。
We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ contain…
Projection Constants and Banach-Mazur Distance in Codimension Two
本文证明余维数一或二且投影常数小于4/3的子空间满足特定Banach-Mazur距离上界，并猜想该上界对所有此类子空间成立。
Taylor spectrum of a Banach module over the quantum plane
本文利用拓扑同调中的横截性关系定义 Banach q-模的联合本质谱与 Taylor 联合谱，并证明其非交换投影 q-性质及缺乏前后向投影性质的例子。
Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications
度量投影算子可以在Hilbert空间和Banach空间中以类似的方式定义。同时，它们的性质也有很大的不同。Hilbert空间中的度量投影算子是一个单调的非扩张算子。它提供了凸闭集的元素对Hilbert空间中任意元素的绝对最佳逼近。这导致了各种各样的应用程序，这个运营商调查理论问题的分析和近似方法。Banach空间中的度量投影算子不具有上述性质，其应用也不简单。度量投影算子方法的两个最重要的应用如下：1。用迭代投影法求解变分不等式，2。用迭代投影法求凸集的公共点。在Banach空间中，这些问题不能在度量投影算子的框架下得到解决。因此，本文在Banach空间中引入新的广义投影算子，作为Hilbert空间中度量投影算子的自然推广。在第2节和第3节中，我们介绍了符号，并回顾了变分不等式理论和逼近理论的一些结果。然后在第四节和第五节中我们讨论了Hilbert空间和Banach空间中度量投影算子$P_Ω$的性质，并利用这些性质建立了变分不等式与直接投影方程之间的等价定理
We investigate properties of essential spectra of disjointness preserving operators acting on Banach $C(K)$-modules. In particular, we prove that under some very mild conditions the upper semi-Fredholm spectrum of such an operator is rotation invariant. In the last part of the paper we provide a full description of the spectrum and the essential spectra of operators acting on Kaplansky modules of the form $T = wU$, where $w \in C(K)$, $U$ is a $d$-isomorphism, and the spectrum of $U$ is a subset of the unit circle.
Projection schemes and the fixed point property in nonreflexive Banach spaces
该研究引入Banach空间投影格式的抽象框架，得到非自反Banach空间满足不动点性质的通用判据，还刻画了变指数Lebesgue空间的不动点性质，给出首个具有该性质的非自反Banach函数格例子。
Boundedness of the averaging projections in nonlocally convex Lorentz sequence spaces and applications to basis theory
研究对称Schauder基在准巴拿赫空间中平均投影的有界性，发现非局部凸情况下该性质不成立，但某些加权洛伦兹序列空间的canonical基具有统一有界平均投影，推翻了局部凸性由有界平均投影决定的假设，并在非局部凸空间中得到新的基结构结论。
Based on the two decades old celebrated Paulsen problem and its solutions for Hilbert spaces by Kwok, Lau, Lee, Ramachandran, Hamilton, and Moitra, we formulate Paulsen problem for Banach spaces. We also formulate projection problem for Banach spaces.
Linearly ordered compacta and Banach spaces with a projectional resolution of the identity
本文构造了一个权为1的紧线性序空间$K$，使得该空间$C(K)$不同构于一个具有单位元的投影分解的巴拿赫空间，而另一方面，$K$是Valdivia紧空间的一个连续像，并且$C(K)$的每个可分子空间都包含在一个1-补可分子空间中.这回答了O. Kalenda和V. Montesinos。
Spectrum of Weighted Composition Operators. Part III. Essential spectra of some disjointness preserving operators on Banach lattices
我们描述基本的（特别是Fredholm和半Fredholm）频谱的巴拿赫格上的形式$T=wU$，其中$w$是一个中心运营商和$U$是一个不相交保持运营商，使其频谱$σ(U)$是一个子集的单位圆。
Given a symbol $φ,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_φ(f)=f\circφ$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $φ$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
Wickstead's conjecture on positive projections and non-representable Banach lattice algebras
本文证明了Wickstead关于正投影的猜想普遍成立，进而解决巴拿赫格代数的表示问题，还在Lean 4中形式化了相关结果。
Band projections in spaces of regular operators
该研究在Dedekind完备Banach格的正则算子空间中引入内带投影，给出原子序连续Banach格的新刻画，并明确了乘法算子成为带投影的充要条件。
We prove that in a large class of Banach spaces of analytic functions in the unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with $G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weighted composition operators if and only if it generates a $C_0$-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$…
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and König, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication ope…
In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the …
We establish a relation between Lipschitz operator ideals and linear operator ideals, which fits in the framework of Galois connection between lattices. We use this relationship to give a criterion which allow us to recognize when a Banach Lipschitz operator ideal is of composition type or not. Also, we introduce the concept of minimal Banach Lipschitz operator ideal, which have analogous properties to minimal Banach operator ideals. Also we characterize minimal Banach Lipschitz operator ideals which are of composition type and present examples which are not of this class.
The metric geometry of subspaces and convex cones of the Banach space revisited
本文证明了在一致凸和一致光滑的Banach空间中，子空间的度量投影线性与凸锥极集的凸性等价，并等价于空间为内积空间。
Phillips' Lemma for L-embedded Banach spaces
本文证明了Phillips引理的下列形式。一个L-嵌入空间的L-投影是弱-弱序列连续的，这是一个巴拿赫空间，它在其双对偶中是可补的，使得两个互补子空间之间的范数是可加的。
Quantified asymptotic behaviour of Banach space operators and applications to iterative projection methods
本文在反射Banach空间中提出了一种算子轨道的渐进行为一般结果，应用于近似理论中的迭代投影方法，如Douglas-Rachford分裂方法和交替投影方法。
The aim of this note is to provide several variants of the diameter two properties for Banach spaces. We study such properties looking for the abundance of diametral points, which holds in the setting of Banach spaces with the Daugavet property, for example, and we intro- duce the diametral diameter two properties in Banach spaces, showing for these new properties stability results, inheritance to subspaces and characterizations in terms of finite rank projections.
In this paper, the split common null point problem in two Banach spaces is considered. Then, using the generalized resolvents of maximal monotone operators and the generalized projections and an infinite family of nonexpansive mappings, a strong convergence theorem for finding a solution of the split common null point problem in two Banach spaces in the presence of a sequence of errors will be proved.
Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems
本文研究流形上光滑函数空间嵌入的拟Banach空间中复合算子的有界性，结合动力系统理论证明仅仿射映射可在单变量整函数拟Banach空间诱导有界复合算子，且二维复仿射空间中非仿射多项式自同构无法诱导该类有界算子。
Let $A$ be a unital Banach algebra over $\mathbb C$, and suppose that the nonzero spectral values of, respectively, $a\mbox{ and }b\in A$ are discrete sets which cluster at $0\in\mathbb C$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, it is further shown that $a$ and $b$ are quasinilpotent equivalent if and only if all the Riesz projec…
Banach lattices of positively homogeneous functions induced by a Banach space
本文研究了由Banach空间诱导的正齐函数Banach空间，探讨了其格同态性质及空间之间的关系，揭示了Banach空间有限维和反射性的特征化条件。
Let $B(Ω)$ be the Banach space of holomorphic functions on a bounded connected domain $Ω$ in $\mathbb C^n$, which contains the ring of polynomials on $Ω$. In this paper, we first establish a criterion for $B(Ω)$ to be reflexive via evaluation functions on $B(Ω)$, that is, $B(Ω)$ is reflexive if and only if the evaluation functions span the dual spaces $(B(Ω))^{*} $. Moreover, under suitable assumptions on $Ω$ and $B(Ω)$, we establish a characterization of the composition operator $C_φ$ to be a Fredholm operator on $B(Ω)$ via the property of the holomorphic self-map $φ:Ω\toΩ$. Our new approach …
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