arXiv subjects

Bentuo Zheng

Publications and source records attributed to Bentuo Zheng.

6 recordsLinked to original sources

arXiv2026

Ball Covering Property on Operators and Calkin Algebra

A Banach space $X$ is said to have the ball covering property (BCP) if the unit sphere of $X$ can be covered by countably many open balls $B(x_i, r_i)$ with $r_i\leq \|x_i\|$ for each $i\in\mathbb{N}$. If there are $R, \delta>0$ so that $r_i\leq R$ and $\|x_i\|-r_i>\delta$ for all $i\in\mathbb{N}$, then we say that $X$ has the uniform ball covering property (UBCP). In this paper, we show that if $X$ has an $1$-unconditional basis or $X$ is an $1$-complemented subspace of a Banach space with a shrinking $1$-unconditional basis, then the Calkin algebra $\mathcal{B}(X)/\mathcal{K}(X)$ fails the BCP. It is also shown that if $X$ has a shrinking unconditional basis with unconditional constant less than 2, then $\mathcal{B}(X)$ has the UBCP.

math.FA↗

arXiv2021

Ball covering property from commutative function spaces to non-commutative spaces of operators

A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let $K$ be a locally compact Hausdorff space and $X$ be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space $C_0(K)$ has the (uniform) BCP if and only if $K$ has a countable $π$-basis. Moreover, we give the stability theorem: the vector-valued continuous function space $C_0(K,X)$ has the (strong or uniform) BCP if and only if $K$ has a countable $π$-basis and $X$ has the (strong or uniform) BCP. We also explore more examples for BCP on non-commutative spaces of operators $B(X,Y)$. In particular, these results imply that $B(c_0)$, $B(\ell_1)$ and every subspaces containing finite rank operators in $B(\ell_p)$ for $1< p<\infty$ all have the BCP, and $B(L_1[0,1])$ fails the BCP. Using those characterizations and results, we show that BCP is not hereditary for 1-complemented subspaces (even for completely 1-complemented subspaces in operator space sense) by constructing two different counterexamples.

math.FA↗

arXiv2014

Wavelets and Triebel type oscillation spaces

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $\dot{F}^{γ_1,γ_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a characterization of $\dot{F}^{γ_1,γ_2}_{p,q}(\mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calderón-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.

math.CA↗

arXiv2013

Perturbations of frames

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.

math.FA↗

arXiv2010

A characterization of Schauder frames which are near-Schauder bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of $c_0$. In particular, a Schauder frame of a Banach space with no copy of $c_0$ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of $c_0$. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.

math.FA↗

arXiv2007

A Characterization of Subspaces and Quotients of Reflexive Banach Spaces with Unconditional Bases

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. Then we prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite dimensional decomposition embeds into a reflexive Banach space with an unconditional basis.

math.FA↗