Reflexivity of the isometry group of some classical spaces

Commutants and Reflexivity of Multiplication tuples on Vector-valued Reproducing Kernel Hilbert Spaces

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…

The principle of local reflexivity and an extension of the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$

By using the Principle of Local Reflexivity (PLR), we prove that for every two Banach spaces $E$ and $X$ there exists a suitable ultrafilter $\mathcal{U}$ such that $ \mathcal{F}(E,X)^*,$ the dual space of the finite rank operators, can be isomorphically identified with certain quotient of the ultrapower space $(E\widehat{\otimes} X^*)_\mathcal{U}$, of the projective tensor product space $E\widehat{\otimes} X^*.$ This generalizes the identity $\mathcal B(E,X^{**})\cong\mathcal B(E,X)^{**}$, where $E$ is finite-dimensional. We then serve our main result to improve some results on the reflexivit…

Large scale geometry of metrisable groups

We develop a theory of large scale geometry of metrisable topological groups that, in a significant number of cases, allows one to define and identify a unique quasi-isometry type intrinsic to the topological group. Moreover, this quasi-isometry type coincides with the classical notion in the case of compactly generated locally compact groups and, for the additive group of a Banach space, is simply that of the corresponding Banach space. In particular, we characterise the class of separable metrisable groups admitting metrically proper, respectively, maximal compatible left-invariant metrics. …

Bargmann transfoms associated with reproducing kernel Hilbert space and application to Dirichlet spaces

The aim of the present paper is three folds. For a reproducing kernel Hilbert space $\mathcal{A}$ (R.K.H.S) and a $σ-$finite measure space $(M_{1},dμ_{1})$ for which the corresponding $L^{2}-$space is a separable Hilbert space, we first build an isometry of Bargmann type as an integral transform from $L^{2}(M_{1},dμ_{1})$ into $\mathcal{A}$. Secondly, in the case where there exists a $σ-$finite measure space $(M_{2},dμ_{2})$ such that the Hilbert space $L^{2}(M_{2},dμ_{2})$ is separable and $\mathcal{A}\subset L^{2}(M_{2},dμ_{2})$ the inverse isometry is also given in an explicit form as an in…

G-强次可微性及其在范数可达子空间中的应用

G-strong subdifferentiability and applications to norm attaining subspaces

本文在群不变映射框架下研究自反性与强次可微性，证明对偶范数G-强次可微蕴含Banach空间G-自反，并推广多个经典泛函分析定理，最终给出范数可达泛函集合中存在闭Banach子空间的充分条件。

Closed Ideals of Operators between the Classical Sequence Spaces

We prove that the spaces $\mathcal L(\ell_p,\mathrm{c}_0)$, $\mathcal L(\ell_p,\ell_\infty)$ and $\mathcal L(\ell_1,\ell_q)$ of operators with $1<p,q<\infty$ have continuum many closed ideals. This extends and improves earlier works by Schlumprecht and Zsák, by Wallis, and by Sirotkin and Wallis. Several open problems remain. Key to our construction of closed ideals are matrices with the Restricted Isometry Property that come from Compressed Sensing.

Some nonlinear characterizations of reflexive Banach spaces

It is well known that in the calculus of variations and in optimization there exist many formulations of the fundamental propositions on the attainment of the infima of sequentially weakly lower semicontinuous coercive functions on reflexive Banach spaces. By either some constructive skills or the regularization skill by inf--convolutions we show in this paper that all these formulations together with their important variants are equivalent to each other and equivalent to the reflexivity of the underlying space. Motivated by this research, we also give a characterization for a normed space to …

Existence and rigidity of quantum isometry groups for compact metric spaces

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.

Bourgain-Morrey序列空间：结构性质、与经典ℓ^p空间的关系及对偶性

Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality

本文研究了离散Bourgain-Morrey序列空间的结构性质、与经典ℓ^p空间的关系及对偶性，揭示了其可分性、嵌入关系及对偶空间的构造。

关于Schatten p-类（0<p<1）的满射最大相位等距映射

On Surjective Max-Phase Isometries of Schatten p-Classes (0<p<1)

本文研究0<p<1的Schatten p-类上的满射最大相位等距映射，在拟巴拿赫空间框架下建立其分类结果，扩展了经典Schatten等距映射分类的适用范围，证明了相关保秩等性质。

The structure of an isometric tuple

An $n$-tuple of operators $(V_1,...,V_n)$ acting on a Hilbert space $H$ is said to be isometric if the operator $[V_1\...\ V_n]:H^n\to H$ is an isometry. We prove a decomposition for an isometric tuple of operators that generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

On the isometric composition operators on the Bloch space in $\mathbb{C}^n$

Let $φ$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\mathbb{C}^n$. In this work, we show that the composition operator $C_φ: f\mapsto f\circ φ$ is bounded on the Bloch space $\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $φ$ to induce an isometry on $\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_φ$ to be an isometry on $\cal{B}$ when $D$ is a Cartan cla…

Generalized 3-circular projections in some Banach spaces

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…

Light groups of isomorphisms of Banach spaces and invariant LUR renormings

Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.

Banach空间的Polish空间。等距和同构类的复杂性

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

本文研究了可分Banach空间的等距和同构类在Banach空间Polish空间中的复杂性，证明了无限维可分Hilbert空间的等距类和同构类的特性，并探讨了其他自然Banach空间类的复杂性。

关于一类新型分数阶Caputo导数Orlicz空间

On a novel fractional Caputo-derivative Orlicz space

本文引入分数阶Caputo导数Orlicz空间，研究其自反性、完备性、可分性及连续嵌入性质，并利用山路定理证明一类分数阶问题存在非平凡弱解。

马祖尔-乌拉姆定理的概率泛化

A Probabilistic Generalization of the Mazur-Ulam Theorem

本文将马祖尔-乌拉姆定理泛化到概率空间，证明几乎处处保持距离的映射几乎处处与全局欧几里得等距映射一致。

A Mazur--Ulam theorem in non-Archimedean normed spaces

The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces.

Orthogonal decompositions and twisted isometries

Let $n > 1$. Let $\{U_{ij}\}_{1 \leq i < j \leq n}$ be $\binom{n}{2}$ commuting unitaries on some Hilbert space $\mathcal{H}$, and suppose $U_{ji} := U_{ij}^*$, $1 \leq i < j \leq n$. An $n$-tuple of isometries $V = (V_1, \ldots ,V_n)$ on $\mathcal{H}$ is called $\mathcal{U}_n$-twisted isometry with respect to $\{U_{ij}\}_{i<j}$ (or simply $\mathcal{U}_n$-twisted isometry if $\{U_{ij}\}_{i<j}$ is clear from the context) if $V_i$'s are in the commutator $\{U_{st}: s \neq t\}'$, and $V_i^*V_j=U_{ij}^*V_jV_i^*$, $i \neq j$ We prove that each $\mathcal{U}_n$-twisted isometry admits a von Neumann-W…

量子比特态空间上的量子Wasserstein等距映射

Quantum Wasserstein isometries on the qubit state space

本文研究量子比特态空间上量子Wasserstein距离的等距映射，证明Pauli矩阵成本下的等距群为酉或反酉共轭并对应O(3)，同时发现clock和shift算子成本下存在非满射、非单射的非常规等距映射。

凸函数、共轭及次微分的Γ收敛

Duality for the $Γ$-convergence of convex functions

本文扩展了凸下连续函数Γ收敛的对偶原理，将其从分离的反射Banach空间推广到更广泛的弱紧生成Banach空间，证明了Γ收敛与共轭函数的Γ收敛在不同拓扑下的等价性。

A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules

Let H be a Hilbert $C^*$-module over a matrix algebra A. It is proved that any function $T:H\to H$ which preserves the absolute value of the (generalized) inner product is of the form $Tf=ϕ(f)Uf$ $(f\in H)$, where $ϕ$ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner's classical unitary-antiunitary theorem for Hilbert modules.

Isomorphisms of Algebras of Convolution Operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that …

Displaying Polish groups on separable Banach spaces

A display of a topological group G on a Banach space X is a topological isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm ||.|| on X, where the latter group is equipped with the strong operator topology. Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has a…

齐次型空间上的Banach格中的紧集

Compact sets in Banach lattices over spaces of homogeneous type

本文证明了齐次型空间上Banach格的Kolmogorov-Riesz紧性定理，用有界性、紧性和平均算子逼近刻画紧性，无需Vitali覆盖等假设，并推广至重排不变空间和Morrey空间。