Florence Lancien, Beata Randrianantoanina, Eric Ricard
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.
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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。
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.
It is shown how any contractive multiplier from the Hardy space to a weighted Hardy space $H^{2}_{\bbeta}$ can be factored as a fixed factor composed with the classical Schur multiplier (contractive multiplier between Hardy spaces). The result is applied to get results on interpolation for a Hardy-to-weighted-Hardy contractive multiplier class.
The optimal constants of the $m$-linear Bohnenblust-Hille and Hardy-Littlewood inequalities are still not known despite its importance in several fields of Mathematics. For the Bohnenblust-Hille inequality and real scalars it is well-known that the optimal constants are not contractive. In this note, among other results, we show that if we consider sums over $M:=M(m)$ indexes with $M\log M=o(m)$, the optimal constants are contractive. For instance, we can consider% \[ M=\left\lfloor \frac{m}{\left( \log m\right) ^{1+\frac{1}{\log\log\log m}}% }\right\rfloor \] where $\lfloor x\rfloor:=\max\{n\…
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.
Fixed Point Theorems for Hardy--Rogers Type Multivalued Mappings in b-Metric Spaces
本文在完备b-度量空间中，通过引入距离依赖控制函数与p阶迭代，改进推广了Chaib等人的Hardy-Rogers型不动点定理，给出p-Hardy-Rogers型多值映射的存在唯一性结论并附实例验证。
We prove that there is no 1-complemented subspace of finite codimension in separable rearrangament-invariant function spaces.
We study Hardy spaces $H^p_ν$ of the conjugate Beltrami equation $\bar{\partial} f=ν\bar{\partial f}$ over Dini-smooth finitely connected domains, for real contractive $ν\in W^{1,r}$ with $r>2$, in the range $r/(r-1)<p<\infty$. We develop a theory of conjugate functions and apply it to solve Dirichlet and Neumann problems for the conductivity equation $\nabla.(σ\nabla u)=0$ where $σ=(1-ν)/(1+ν)$. In particular situations, we also consider some density properties of traces of solutions together with boundary approximation issues.
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.
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.
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.
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。
On projection from $A^p_ω$ to the Hardy spaces $H^p$
本文综述了从一般全纯空间 $A^p_ω$ 到 Hardy 空间 $H^p$ 的投影已知结果，并给出了上半平面情形的新投影结果。
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。
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。
Contractive Hardy--Littlewood inequalities in the Dirichlet range
本文研究 Dirichlet 范围（0<α<1）内的解析函数空间 A_α^p，证明压缩不等式 ||f||_{β,q} ≤ ||f||_{α,p} 在 α/p=β/q 时成立，并确定等号条件，推广了经典 Bergman 空间的结果。
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…
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。
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.
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…
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…
Metric projections, zeros of optimal polynomial approximants, and some extremal problems in Hardy spaces
本文研究了Hardy空间H^p中单位常数函数在移不变子空间上的度量投影问题，揭示了最佳逼近与极值问题的关系，并探讨了相关不等式及子空间结构。
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…
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.
A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operator topology. In this article, we prove that for contractions $T$, which commute with certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued m…
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。
Embedding and compact embedding between Bergman and Hardy spaces
本文确定了${\rm{\bf C}}^n$开单位球上Hardy空间与加权Bergman空间间嵌入及紧嵌入的精确条件，覆盖所有可能情况，引入“紧拟合”概念并提出相关猜想。
For a singular probability measure $μ$ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to $μ$. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szegő kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of $μ$. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are …
Sub-Hardy Hilbert spaces in the non-commutative unit row-ball
本文将经典 Hardy 空间理论中的 Szegö 定理与 Toeplitz 算子分解等思想推广至全 Fock 空间的多变量非交换设定，并证明了非交换有理函数的 Fejér-Riesz 型定理。
We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirich…
Projections and minimal invariant subspaces in the Hardy space over the bidisk
本文研究双圆盘Hardy空间中正交投影的刚性，证明极小不变子空间的非零投影具有共同的$t_z^*$-不变结构，并关联到不变子空间问题。
Products of Siegel upper half spaces are Siegel domains, whose Silov boundaries have the structure of products $\mathscr H_1\times\mathscr H_2$ of Heisenberg groups. By the reproducing formula of bi-parameter heat kernel associated to sub-Laplacians, we show that a function in holomorphic Hardy space $H^1$ on such a domain has boundary value belonging to bi-parameter Hardy space $ H^1 (\mathscr H_1\times \mathscr H_2)$. With the help of atomic decomposition of $ H^1 (\mathscr H_1\times \mathscr H_2)$ and bi-paramete rharmonic analysis, we show that the Cauchy-Szeg\H o projection is a bounded o…
A non-commutative de Branges-Rovnyak model for row contractions
本文提出非交换德布兰-罗文纳克模型，用于描述CNC行收缩的特征函数及其单位不变性。
The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then stu…
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 study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…
The full Fock space over $\mathbb C ^d$ can be identified with the free Hardy space, $H^2 (\mathbb B ^d _\mathbb N)$ - the unique non-commutative reproducing kernel Hilbert space corresponding to a non-commutative Szegö kernel on the non-commutative, multi-variable open unit ball $\mathbb B ^d _\mathbb N := \bigsqcup _{n=1} ^\infty \left( \mathbb C^{n\times n} \otimes \mathbb C ^d \right) _1$. Elements of this space are free or non-commutative functions on $\mathbb B ^d _\mathbb N$. Under this identification, the full Fock space is the canonical non-commutative and several-variable analogue of…
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 two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz bas…
Diagonal operators on Janson-Sobolev and Janson-Sobolev-Hardy spaces
本文研究 Janson-Sobolev 与 Janson-Sobolev-Hardy 空间上的有界对角算子，证明其可被相关算子以常数 $1^+$ 几乎投影地分解，并给出在自然有界性条件及 $L^1$-范数下的分解推论。
Let $H[X]$ and $H[Y]$ be abstract Hardy spaces built upon Banach function spaces $X$ and $Y$ over the unit circle $\mathbb{T}$. We prove an analogue of the Brown-Halmos theorem for Toeplitz operators $T_a$ acting from $H[X]$ to $H[Y]$ under the only assumption that the space $X$ is separable and the Riesz projection $P$ is bounded on the space $Y$. We specify our results to the case of variable Lebesgue spaces $X=L^{p(\cdot)}$ and $Y=L^{q(\cdot)}$ and to the case of Lorentz spaces $X=Y=L^{p,q}(w)$, $1<p<\infty$, $1\le q<\infty$ with Muckenhoupt weights $w\in A_p(\mathbb{T})$.
de Branges-Rovnyak spaces which are complete Nevanlinna-Pick spaces
本文刻画了 de Branges-Rovnyak 空间为完全 Nevanlinna-Pick 空间的条件，推广了 Hardy 空间和 Drury-Arveson 空间的已有结果，并证明双圆盘或单位圆盘情形仅当等距同构于 Hardy 空间时成立，而 $n\ge 3$ 时永不成立。
Hardy spaces of discrete holomorphic functions on the upper half-lattice
该研究在正方形格点离散全纯性框架下，发展上半格离散全纯函数的Hardy空间\(H^p\)理论，证明相关公式与恒等式，描述再生核等，比较离散与经典理论，还证明对偶性等结果。
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.
Motivated by Barr{\'ı}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. T…
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。
Weighted Hardy-Rellich inequalities via the Emden-Fowler transform
该研究利用埃姆登-福勒变换证明锥上（含穿孔空间、半空间等特例）的最优哈代-雷利不等式，确定测试函数类的最优常数，分析极值不存在性、共振现象及紧支集测试函数的排除情况，并给出带最优余项的改进不等式。
In this paper we give sufficient conditions on $α\geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^α)Δu+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image…
In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obt…
For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not…
Contractivity of Rudin's diagonal restriction operator for $H^p$ of the polydisc
该研究证明了Rudin对角限制算子在$2\leq p<\infty$时从双圆盘$H^p$到圆盘$A^p$的压缩性，刻画了等号情形，推广到多圆柱并获多项式点赋值等应用，证明依赖于格林公式推导的范数幂差显式公式。
On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces
本文证明了在建立在 Banach 函数空间上的抽象 Hardy 空间中，Toeplitz 算子的本质范数等于函数的 $L^\infty$ 范数的充要条件是后向移位算子的本质范数为 1，推广了经典 Hardy 空间的相关结论。
We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of $\mathcal{H}^p$. More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy--Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$, contrasting the usual $L^p$ duality. We next deduce general coe…
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…
Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces
本文研究了用Lebesgue $p$-范数估计Riesz投影算子表达式的精确不等式，改进了已有结果并证明了相关猜想，同时导出了多种函数空间上的Riesz型定理。
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