Beata Randrianantoanina
This is the survey of results about norm one projections and 1-complemented subspaces in Köthe function spaces and Banach sequence spaces. The historical development of the theory is presented from the 1930's to the newest ideas. Proofs of the main results are outlined. Open problems are also discussed. Every effort has been made to include as complete a bibliography as possible.
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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…
We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X)$, with $\Om$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.
A Strong Convergence Theorem for finite families of Bregman Demimetric Mappings in a Banach Space under a New Shrinking projection Method
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.
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.
In this paper, a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space is proved by using metric projections. This theorem extends and improves the recent strong convergence theorem due to Matsushita and Takahashi [ Appl. Math. Comput. 196 (2008) 422-425] which was established for nonexpansive mappings.
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。
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…
Strong convergence of a new iterative process based on the Shrinking projection method to a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the solution set of an equilibrium problem in a Banach space is established. Our results improved and extend the corresponding results announced by many others.
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。
In the article is shown that classes of finite equivalence that are generated by Lebesgue-Riesz spaces Lp have the amalgamation property if and only if p is not equal to an even natural number that is strongly large then 2.
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。
Recently in a series of papers it is observed that in many Banach spaces, which include classical spaces $C(Ω)$ and $L_p$-spaces, $1 \leq p < \infty, p \neq 2$, any generalized bi-circular projection $P$ is given by $P = \frac{I+T}{2}$, where $I$ is the identity operator of the space and $T$ is a reflection, that is, $T$ is a surjective isometry with $T^2 = I$. For surjective isometries of order $n \geq 3$, the corresponding notion of projection is generalized $n$-circular projection as defined in \cite{AD}. In this paper we show that in a Banach space $X$, if generalized bi-circular projectio…
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。
According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof of convergence is based on a known criterion in terms of the boundary spectrum.
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…
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。
This paper is an account (without proofs) of the results of our work "Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications", funct-an/9311001. The Section 9 establishing a connection between variational inequalities and Wienner-Hopf equations in Banach spaces by means of metric and generalized projection operators, is added.
A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of kno…
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…
In 1979, B.Bjornestal obtained local estimate for a modulus of uniform continuity of metric projection operator on closed subspace in uniformly convex and uniformly smooth Banach space $B$. In the present paper we give the global version of this result for the projection operator on an arbitrary closed convex set in $B$.
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.
Previously unknown estimates of uniform continuity of projection operators in Banach space have been obtained. They can be used in the investigations of approximation methods, in particular, the method of quasisolutions, methods of regularization and penalty functions, for solving nonlinear problems on exact and peturbed sets.
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 projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It provides an absolutely best approximation for arbitrary elements from Hilbert space by the elements of convex closed sets . This leads to a variety of applications of this operator for investigating theoretical questions in analysis and for approximation methods. Metric projection operators in Banach space do not have properties mentioned above and their ap…
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.
We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.
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。
Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators.
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 show that if $\mathfrak{A}$ is a commutative complex non-unital Banach Algebra with norm $\|\cdot\|$, then $\|\cdot\|$ is regular on $\mathfrak{A}$ if and only if $\|\cdot\|_{op}$ is a norm on $\mathfrak{A}\oplus \mathbb{C}$ and $\mathfrak{A}\oplus\mathbb{C}$ is a commutative complex Banach Algebra with respect to $\|\cdot\|_{op}$.
Norms and Non-Equivalence in Infinite-Dimensional Banach Spaces
本文研究无穷维巴拿赫空间中不同范数的相互作用，分析双射线性映射诱导的范数、非等价范数的存在性以及哈默尔基的作用。
Mordukhovich derivatives of the normalized duality mapping in Banach spaces
本文研究了Banach空间中规范化对偶映射的Mordukhovich导数的性质，考察了一致凸且一致光滑的Banach空间lp以及一般Banach空间L1和C[0,1]三种情形。
We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the al…
Equivalent quasi-norms on rearrangement-invariant quasi-Banach function spaces
本文针对重排不变拟Banach函数空间，提出了用于确定其等价拟范数的新公式
Let $X$ be a complex Banach space; in this work we characterize the property of Frechet diffrentiability for the dual space $X^*$.
We introduce and study the enveloping norms of regularly P-operators between Banach lattices E and F, where P is a subspace of the space L(E,F) of continuous operators from E to F. We prove that if P is closed in L(E,F) in the operator norm then the regularly P-operators forms a Banach space under the enveloping norm. Conditions providing that regularly P-operators forms a Banach lattice under the enveloping norm are given.
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 discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.
In this note, we provide a starlike and normal tiling in any separable Banach space. That means, there are positive constants r and R (not depending on the separable Banach space) such that every tile of this tiling is starlike, contains a ball of radius r and is contained in a ball of radius R.
We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.
Necessary and sufficient conditions for Banach space to be(isometrically isomorphic to) a dual space will be given.
In this paper, we study the connections between the normality, regularity, full regularity, and chain-complete property in partially ordered Banach spaces. Then, by applying these properties, we prove some fixed point theorems on partially ordered Banach spaces. As applications of these fixed point theorems, we prove the existence of solutions of some integral equations, such as Hammerstein integral equations, in Banach spaces.
In this paper, we study octahedral norms in free Banach lattices $FBL[E]$ generated by a Banach space $E$. We prove that if $E$ is an $L_1(μ)$-space, a predual of von Neumann algebra, a predual of a JBW$^*$-triple, the dual of an $M$-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of $FBL[E]$ is octahedral. We get the analogous result when the topological dual $E^*$ of $E$ is almost square. We finish the paper by proving that the norm of the free Banach lattice generated by a Banach space of dimension $ \geq 2$ is nowhere Fréchet di…
We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter 2, which answer an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combinat…
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。
In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space - that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive - is weak-weakly sequentially continuous.
In this paper we collect results concerning the {operator-norm} convergent {Trotter} product formula for two semigroups $\{\e^{- t A}\}_{t\geq 0}$, $\{\e^{- t B}\}_{t\geq 0}$, with densely defined generators $A$ and $B$ in a {Banach} space. Although the {strong} convergence in Banach space for contraction semigroups is known since the seminal paper by Trotter (1959), which after more than three decades was extended to convergence in the {operator-norm} topology in {Hilbert} spaces by Rogava (1993), the {operator-norm} convergence in a {Banach} space was established only in (2001).
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。
Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided.
The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping.
Given a finite measure space $(Ω,Σ,μ)$, we show that any Banach space $X(μ)$ consisting of (equivalence classes of) real measurable functions defined on $Ω$ such that $f χ_A \in X(μ) $ and $ \|f χ_A \| \leq \|f\|, \, f \in X(μ), \ A \in Σ$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.
We study the smoothness and the norm attainment of bounded bilinear operators between Banach spaces, using the concepts of Birkhoff-James orthogonality and semi-inner-products. In the finite-dimensional case, we characterize Birkhoff-James orthogonality of bilinear operators in terms of the norm attainment set. This yields a nice characterization of smoothness of bilinear operators between Banach spaces, in the finite-dimensional case. Without any restriction on the dimension of the space, we obtain a complete characterization of the norm attainment set of a bounded bilinear operator using sem…
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