A twisted $\bar{\partial}_f$-Neumann problem and Toeplitz $n$-tuples from singularity theory

单位欧几里得球中3×3矩阵的冯·诺依曼不等式

The von Neumann inequality for $3\times3$ matrices in the unit Euclidean ball

本文证明交换且行收缩的3×3矩阵d元组的冯·诺依曼不等式常数与d无关，并给出其数值估计。

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von …

A tuple of multiplication operators defined by twisted holomorphic proper maps

This paper mainly concerns the von Neumann algebras induced by a tuple of multiplication operators on Bergman spaces which arise essentially from holomorphic proper maps over higher dimensional domains. We study the structures and abelian properties of the related von Neumann algebras, and in interesting cases they turns out to be tightly related to a Riemann manifold. There is a close interplay between operator theory, geometry and complex analysis. Many examples are presented.

Isometric dilations and von Neumann inequality for finite rank commuting contractions

Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We always assume that $n \geq 3$. The importance of this class of $n$-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for $n$-tuples, $n \geq 3$, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, w…

Orthogonal decompositions and twisted isometries II

We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify al…

行压缩矩阵元组的冯·诺依曼不等式

von Neumann's inequality for row contractive matrix tuples

本文证明行压缩矩阵元组的冯·诺依曼不等式（含常数$C_n$），并将其应用于解决多个开放问题，包括Gleason问题的不可压缩求解性、加权狄利克雷空间乘子代数的拓扑次齐次性及相关代数表示的确定等。