arXiv subjects

Duanxu Dai

Publications and source records attributed to Duanxu Dai.

5 recordsLinked to original sources

arXiv2018

On Qian's problem for $\mathcal{L}_{\infty}$-spaces

In this paper we devote to study Qian's problem for $\mathcal{L}_{\infty}$-spaces. Firstly, a positive answer to Qian's problem for $C(K)$-spaces is given by the assumption that $K$ has the C$\check{e}$ch-Stone property. Secondly, we obtain quantitative characterizations of separably injective spaces that turn out to give a positive answer to Qian's problem of 1995 in the setting of separable universality. Thirdly, we prove a sharpen quantitative and generalized Sobczyk theorem, which gives sharpen constants ($α,γ$) for Qian's Problem. Finally, we give a more generalized Figiel theorem for $\mathcal{L}_{\infty}$-spaces.

math.FA↗

arXiv2017

Subdifferential representation of convex functions on $X^*$

In this paper, we obtain subdifferential representation of a proper $w^*$-lower semicontinous convex function on $X^*$ as follows: Let $g$ be a proper convex $w^*$-lower semicontinuous function on $X^*$. Assume that int dom $g$ $\neq\emptyset$ (resp. int (dom ($g^*|_X)$)$\neq\emptyset$). Then given any point $x_0^*$ $\in$ D ($\partial g\cap X$) and $x^*$ $\in$ dom $g$ (resp. $x^*\in X^*$), we have $$g(x^*)=g(x_0^*)+\sup\{\sum_{i=0}^{n-1}\langle x_i,x_{i+1}^*-x_i^*\rangle +\langle x_n,x^*-x_n^*\rangle \},$$ where the above supremum is taken over all integers $n$, all $x_i^*\in X^*$ and all $x_i\in\partial g(x_i^*)\cap X$ for $i=0,1,\cdots,n$. (resp. if, moreover, $X^*$ has the Radon-Nikodym property, then we may estimate the above supremum among the set of $w^*$-strongly exposed points of $g$.)

math.FA↗

arXiv2014

Stability of Banach spaces via nonlinear $\varepsilon$-isometries

In this paper, we prove that the existence of an $\varepsilon$-isometry from a separable Banach space $X$ into $Y$ (the James space or a reflexive space) implies the existence of a linear isometry from $X$ into $Y$. Then we present a set valued mapping version lemma on non-surjective $\varepsilon$-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by $\varepsilon$-isometries.

math.FA↗

arXiv2013

Universal stability of Banach spaces for $\varepsilon$-isometries

Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard $\varepsilon$-isometry $f:X\rightarrow Y$ is said to be $(α,γ)$-stable (with respect to $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ for some $α, γ>0$) if $T$ is a linear operator with $\|T\|\leqα$ so that $Tf-Id$ is uniformly bounded by $γ\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(α,γ)$-stable for some $α,γ>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that universal right-stability spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space $X$ isomorphic to a subspace of $\ell_\infty$ is universally left-stable if and only if it is isomorphic to $\ell_\infty$; and that a separable space $X$ satisfies the condition that $(X,Y)$ is left-stable for every separable $Y$ if and only if it is isomorphic to $c_0$.

math.FA↗

arXiv2013

On universal left-stability of $ε$-isometries

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $γ>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ such that $\|Tf(x)-x\|\leqγ\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that if a dual Banach space $X$ is universally-left-stable, then it is isometric to a complemented $w^*$-closed subspace of $\ell_\infty(Γ)$ for some set $Γ$, hence, an injective space; and that a Banach space is universally-left-stable if and only if it is a cardinality injective space; and universally-left-stability spaces are invariant.

math.FA↗