arXiv subjects

Zhizheng Yu

Publications and source records attributed to Zhizheng Yu.

3 recordsLinked to original sources

arXiv2026

On strengthened versions of Klee's convex body problem in Banach spaces

In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and G\^{a}teaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fr\'{e}chet smoothness and uniform smoothness. As a result, we show (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$ can be approximated by Fr\'echet smooth convex bodies if $X$ admits an equivalent norm so that its dual norm is locally uniformly convex on $X^*$; 3. Every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if $X$ is reflexive; 4. If $X$ is separable, then every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if and only if $X$ is an Asplund space; (5) the following statements are equivalent: A. $X$ is super reflexive; B. Every convex body in $X$ can be uniformly approximated by uniformly convex bodies; C. Every convex body in $X$ can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in $X$ can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies.

math.FA↗

arXiv2026

Arazy-type decomposition theorem for bounded linear operators and commutators on the trace class

The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal $\mathcal C_E$ of the algebra $\mathcal B(H)$ of all bounded linear operators on the separable infinite-dimensional Hilbert space $H$. In this paper, we extend and strengthen Arazy's decomposition theorem to the setting of general bounded linear operators on a separable (quasi-Banach) operator ideal $\mathcal C_E$ of $\mathcal B(H)$. Several applications are given to the study of $\mathcal C_E$-strictly singular operators, largest proper ideals in the algebra $\mathcal B(\mathcal C_E)$ of all bounded linear operators on $\mathcal C_E$ and complementably homogeneous Banach spaces among others. Our versions of decomposition theorems supply tools for a noncommutative generalization of deep commutator theorems for operators on $\ell_p$ and $L_p$, $1\le p <\infty $, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class $\mathcal C_p$, $1\le p<\infty $. For the crucial case, $p=1$, we establish that any operator $T\in\mathcal B(\mathcal C_1)$ is a commutator if and only if $T$ is not of the form $\lambda I+K$ for some $\lambda\neq 0$ and $\mathcal C_1$-strictly singular operator $K$.

math.FA↗

arXiv2024

Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras

Let $\mathcal A$ be a infinite dimensional C*-algebra and $1<p<\infty$. We compute the Szlenk index of $\mathcal A$ and $L_p(\mathcal A)$, and show that $Sz(\mathcal A)=Γ'(i(\mathcal A))$ and $Dz(\mathcal A)=Sz(L_p(\mathcal A))=ωSz(\mathcal A)=ωΓ'(i(\mathcal A))$, where $i(\mathcal A)$ is the noncommutative Cantor-Bendixson index, $Γ'(ξ)$ is the minimum ordinal number which is greater than $ξ$ of the form $ω^ζ$ for some $ζ$ and we agree that $Γ'(\infty)=\infty$ and $ω\cdot\infty=\infty$. As a application, we compute the Szlenk index [respectively, $w^\ast$-dentability index] of a C*-tensor product $\mathcal A\otimes_β\mathcal B$ of non-zero C*-algebras $\mathcal A$ and $\mathcal B$ in terms of $Sz(\mathcal A)$ and $Sz(\mathcal B)$ [respectively, $Dz(\mathcal A)$ and $Dz(\mathcal B)$]. When $\mathcal A$ is a separable C*-algebra, we show that there exists $a\in \mathcal A_h$ such that $Sz(\mathcal A)=Sz(C^\ast(a))$ and $Dz(\mathcal A)=Dz(C^\ast(a))$, where $C^\ast(a)$ is the C*-subalgebra of $\mathcal A$ generated by $a$.

math.FA↗