Beata Randrianantoanina
We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.
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We prove that there is no 1-complemented subspace of finite codimension in separable rearrangament-invariant function spaces.
In this paper, we firstly character the structures of symmetries $J$ such that a projection $P$ is $J$-contractive. Then the minimal and maximal elements of the symmetries $J$ with $P^{\ast}JP\leqslant J$(or $JP\geqslant0)$ are given. Moreover, some formulas between $P_{(2I-P-P^{\ast})^{+}}$ $(P_{(2I-P-P^{\ast})^{-}})$ and $P_{(P+P^{\ast})^-}$ $(P_{(P+P^{\ast})^+})$ are established.
We characterize norm one complemented subspaces of Orlicz sequence spaces $\ell_M$ equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function $M$ is sufficiently smooth and sufficiently different from the square function. This paper concentrates on the more difficult real case, the complex case follows from previously known results.
Extension of contractive projections
本文通过建立扩张定理，给出了Hardy空间中所有收缩投影和1-补子空间的显式表达式，并证明了其非平凡1-补子空间的等距同构性质，回答了Wojtaszczyk在2003年提出的问题。
Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries
该研究否定了扎瓦尔齐娜关于预紧拟度量空间双射非压缩自映射是否为等距映射的问题，构造了反例并完成机器验证，还证明了单值性定理，界定了未解决问题。
Contractive projections, conditional expectations, and idempotent coefficient multipliers on $H^p$ spaces $(0<p<1)$
本文研究了 $0<p<1$ 时 Hardy 空间上的压缩投影、条件期望与幂等系数乘子，通过建立概率 $L^p$ 空间中压缩投影的延拓定理，完整刻画了 $d$ 维环面上此类乘子，补充了现有结果。
Given a JBW*-triple Z and a normal contractive projection P on Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z.
A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $ϕ:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $ϕ$ is similar to a completely contractive map $ϕ'$. Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse $ϕ'^{-1}$ completely contractive as well whenever the map $ϕ$ has a completely bounded inverse. We show that there exist invertible operators $X$ and $Y$ such that the map $$ XaX^{-1}\mapsto Yϕ(a)Y^{-1} $$ is completely contractive and is "almost" isometric on a…
We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which a…
We constructed a unitary semigroup $(e^{tA})_{t \geq 0}$ on a Hilbert space and an orthogonal projection $P$ such that the limit $\lim_{n \to \infty} [ e^{\frac{t}{n}A}P ]^n$ does not exist strongly. A similar example with a positive contractive semigroup and positive contractive projection on $L_p$ is also constructed.
Let $S_1$ and $S_2$ be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets $S_1,S_2$ and exponents $p$ such that the canonical projection $P$ from $PW_{S_1\cup S_2}^p$ to $PW_{S_1}^p$ is a contraction.
We prove a conjecture of Wojtaszczyk that for $1\leq p<\infty$, $p\neq 2$, $H_p(\mathbbT)$ does not admit any norm one projections with dimension of the range finite and bigger than 1. This implies in particular that for $1\leq p<\infty$, $p\ne 2$, $H_p$ does not admit a Schauder basis with constant one.
On the structure of contractively decomposable projections on noncommutative $L^p$-spaces and Schatten spaces
【一句话总结】研究 \(1 < p < \infty\) 时非交换Haagerup \(L^p\) 空间上压缩可分解投影范围的同构情况，在Schatten空间有更精确描述，还证明了Schatten空间上压缩1 - 伪可分解投影与压缩可分解投影的等价性。
Positive contractive projections on noncommutative $\mathrm{L}^p$-spaces and nonassociative $\mathrm{L}^p$-spaces
本文改进了已有研究，精确刻画了与 $σ$-有限冯·诺依曼代数相关的非交换 $\mathrm{L}^p$ 空间上的正压缩投影，并证明其值域在多数情况下与 $\mathrm{JW}^*$-代数相关的非结合 $\mathrm{L}^p$ 空间等距。
We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell…
Botelho, Jamison, and Molnár \cite{BJM}, and Geh\' er and Šemrl \cite{GeS} have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space $H$. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set $P_\infty (H)$ of all projections of infinite rank and infinite corank when $H$ is separable. The proof technique is entirely different from th…
A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces
本文提出基于 Levi 形式的模理论，刻画 Banach 空间压缩投影，并应用于非交换 L^p 空间，证明 2-压缩投影等价于完全压缩，且值域具有三元结构。
Completely isometric subspaces of noncommutative $\mathrm{L}^p$-spaces and contractive projections
该研究针对1<p<∞且p≠2的非交换L^p空间，建立正压缩投影值域完全等距于另一非交换L^p空间时必为完全正投影的结论，还刻画了可压缩分解投影的值域特征。
Parallel to the study of finite dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces H_n^k,0< k < n+1, generalizing the row and column Hilbert spaces R_n,C_n and show that an atomic subspace X of B(H) which is the range of a contractive projection on B(H) is isometrically completely contractive to a direct sum of the H_n^k and Cartan factors of types 1 to 4. In particular, for finite dimensional X, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classific…
In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.
Let X be an L_1-predual space and let K be a countable linearly independent subset of the extreme points of its closed dual ball. It is shown that if the norm-closed linear span Y of K is w^*-closed in X^*, then Y is the range of a w^*-continuous contractive projection in X^*. This result is applied in order to provide new and simpler proofs of the results of Lazar, Lindenstrauss and Zippin on the embedding of C(K) spaces into X.
In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of…
Let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\leq p<\infty$. We say that $T$ is $\ell^1$-bounded (resp. $\ell^1$-contractive) if $T\otimes I_{\ell^1}$ extends to a bounded (resp. contractive) map from $L^p({\mathcal M};\ell^1)$ into $L^p({\mathcal N};\ell^1)$. We show that Yeadon's factorization theorem for $L^p$-isometries, $1\leq p\not=2 <\infty$, applies to an isometry $T\colon L^2({\mathcal M})\to L^2({\mathcal N})$ if and only if $T$ is $\ell^1$-contractive. We also show that a contractive operator $T\colon L^p({\mat…
In this paper we describe the surjective linear isometries on a vector valued little Bloch space with range space a strictly convex and smooth complex Banach space. We also describe the hermitian operators and the generalized bi-circular projections supported by these spaces.
2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces
本文研究非交换L^p空间上的2-正压缩投影问题，证明其值域完全等距序同构于某个非交换L^p空间，该结果对Schatten空间亦为全新结论，方法上本质依赖非迹的Haagerup非交换L^p空间。
Positive contractive projections in Schatten Spaces
本文研究了Schatten空间中正1-补足子空间的特征，基于Arazy和Friedman的工作，确定了五种互不相同的不可分解正1-补足子空间类型，并证明所有正1-补足子空间均可表示为这些不可分解子空间的直和。
We prove that if the group of isometries of C(X,E) is algebraically reflexive, then the group of n-isometries is also algebraically reflexive. Here, X is a compact Hausdorff space and E is a Banach space. As a corollary to this, we establish the algebraic reflexivity of the set of generalized bi-circular projections on C(X,E).
The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.
It is known that the non-zero part of compact defect operators of Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\ & & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators and $D$ is a positive contractive diagonal operator. We discuss the question of constructing an irreducible BCL pair from a diagonal operator of the above type. The answer to this question is sometimes in the affirmative and sometimes in the negative. This also answers a part of the question raised …
The aim of this paper is to show that two accessible subspaces in the Loynes $\mathcal{Z}$ - space $\Hc$ are the initial and final space of a partial gramian isometry, respectively if the norm of the difference of the associated gramian selfadjoint projections is strictly less than 1.
Contractive inequalities between Dirichlet and Hardy spaces
本文证明了 Brevig 等人关于 Dirichlet 空间与 Hardy 空间之间压缩不等式的猜想，并探讨了其与 Riesz 投影的联系。
Botelho, Jamison, and Moln\' ar have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Moln\' ar, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surectivit…
We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.
Isometries and geometric liftings for Alexiewicz-normed $L^\infty$ spaces
本文研究实数轴紧致子集上本质有界函数空间，基于原始函数的上确界范数进行等距映射分类，通过测度投影确定由符号和递增双 Lipschitz 映射决定的加权复合算子，并给出几何条件以判定该区间映射能否提升为紧致集上的同胚或双 Lipschitz 同胚。
Based on the characterization of surjective $L^p$-isometries of unitary groups in finite factors, we describe all surjective $L^p$-isometries between Grassmann spaces of projections with the same trace value in semifinite factors.
On the contractivity of the Riesz projection
本文证明了Riesz投影在$1<q<\infty$时从$L^q$到$L^{4(1-1/q)}$的压缩性，填补了相关猜想未解决的参数范围。
We construct some separable infinite dimensional homogeneous Hilbertian operator spaces which generalize the row and column spaces R and C. We show that separable infinite-dimensional Hilbertian JC*-triples are completely isometric to an element of the set of (infinite) intersections of these spaces. This set includes the operator spaces R, C, their intersection, and the space spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that these new spaces are completely isometric to the space of creation (resp. annihilation) operators on the anti-symmetric tensors o…
A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula $(e^{\frac{t}{n}A}P)^n$ (where $P$ denotes a bounded projection), we prove that whenever the generator $A$ is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is {\it{not}} quasi-contractive.
We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW$^*$-triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW$^*$-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW$^*$-triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projec…
Isometries without linear shadows: failure of Figiel's theorem beyond local convexity
本文证明Figiel定理在非局部凸空间（$0<p<1$）中完全失效，构造了无线性左逆的等距嵌入，并利用Lipschitz-free $p$-空间及定量估计，得出等距$p$-提升性质仅在一维空间成立。
Some results on fixed points related to the contractive compositions of bounded operators in complete metric spaces are discussed through the manuscript. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators.
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the …
In this note we include two remarks about bounded ($\underline{not}$ necessarily contractive) linear projections on a von Neumann-algebra. We show that if $M$ is a von Neumann-subalgebra of $B(H)$ which is complemented in B(H) and isomorphic to $M \otimes M$ then $M$ is injective (or equivalently $M$ is contractively complemented). We do not know how to get rid of the second assumption on $M$. In the second part,we show that any complemented reflexive subspace of a $C^*$- algebra is necessarily linearly isomorphic to a Hilbert space.
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…
Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ are clopen subsets of $M^{-1}$ and $N^{-1}$, respectively, which are closed for powers, inverses and products of the form $U_{a} (b)$. In this paper we prove that for each surjective isometry $Δ: \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $Δ(a) = T_0(a) +u_0$ f…
The closed one-sided ideals of a C*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C*-algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B** which also lies in the weak* closure of A. Although this seems quite natural, the proof requires a set of new techniques which may may be viewed as a noncommutative version o…
Generalized tri-circular projections on some spaces of analytic functions
本文研究开单位圆盘上Hardy、Bergman、Bloch等Banach函数空间及$\u27e8K\u27e9$值解析函数空间$S^p_{\mathcal{K}}$上的广义三圆压缩投影，给出其结构与完整刻画，丰富了解析函数空间的投影理论。
Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend…
We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…
Transfinite Iteration of Operator Transforms and Spectral Projections in Hilbert and Banach Spaces
研究了有界算子变换多层迭代的收敛性，在泛函演算假设下证明了向谱投影的强收敛，并提供了显式滤波器、单调性界及反例。
The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exists a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_Φ, M_Ψ)$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where \[ Φ(z)=(P+zP^{\perp})U^*\;\text{and}\; Ψ(z)=U(P^{\perp}+zP) \;;(…
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