[论文解读] Weighted shifts on directed trees

Zenon Jan Jabłoński, Il Bong Jung+1 more|arXiv (Cornell University)|Nov 19, 2009

Holomorphic and Operator Theory参考文献 41被引用 4

一句话总结

本文引入并研究了无限有向树上的加权移位，作为经典加权移位和加权邻接算子的推广。它通过权重给出了共轭性、共轭共轭性、次正规性和完全超扩张性的完整刻画，关键结果表明：在特定树 ${\mathscr{T}}_{\eta,\kappa}$ 上的次正规性等价于经典单边加权移位的 $k$-步向后可延拓性。

ABSTRACT

A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and complete hyperexpansivity of weighted shifts on them. A strict connection of the latter with $k$-step backward extendibility of subnormal as well as completely hyperexpansive unilateral classical weighted shifts is established. Models of subnormal and completely hyperexpansive weighted shifts on these particular trees are constructed. Various illustrative examples of weighted shifts on directed trees with the prescribed properties are furnished. Many of them are simpler than those previously found on occasion of investigating analogical properties of other classes of operators.

研究动机与目标

• 引入并系统研究无限有向树上的加权移位，作为经典加权移位和加权邻接算子的推广。
• 在此新框架内刻画基本算子理论性质——如闭性、伴随算子、极分解和模——。
• 研究加权移位与其伴随算子之间定义域的关系，特别是在各种包含条件下的关系。
• 通过树上的权重，提供共轭性、共轭共轭性、次正规性和完全超扩张性的完整刻画。
• 在特定有向树上，特别是 ${\mathscr{T}}_{\eta,\kappa}$ 上，构建次正规和完全超扩张算子的模型，并将其与经典单边加权移位的 $k$-步向后可延拓性联系起来。

提出的方法

• 通过边集上的权重函数 $\lambda_v$ 定义有向树上的加权移位，推广经典加权移位和加权邻接算子。
• 利用图论结构——特别是具有一个分支顶点的无限有向树（${\mathscr{T}}_{\eta,\kappa}$）——来建模和分析算子性质。
• 应用谱理论和函数演算研究圆性、弗雷德霍姆性以及定义域包含关系 $\mathscr{D}(S_{\boldsymbol{\lambda}}) \subseteq \mathscr{D}(S_{\boldsymbol{\lambda}}^*)$。
• 采用涉及权重不等式的 $p$-共轭性条件：$|\lambda_{i,j}| \leq |\lambda_{i,j+1}|$，$\left(\sum |\lambda_{i,1}|^2\right)^{p-1} \left(\sum \frac{|\lambda_{i,1}|^2}{|\lambda_{i,2}|^{2p}}\right) \leq 1$，以及对 $\lambda_0$ 的有界条件。
• 建立在 ${\mathscr{T}}_{\eta,\kappa}$ 上的次正规性与经典单边加权移位的 $k$-步向后可延拓性之间的严格联系。
• 使用测度论和矩条件（例如 $\mu_1 = \delta_{1/a^2}$，$\mu_2 = \delta_{1/b^2}$）通过权重参数刻画次正规性。

实验结果

研究问题

• RQ1 在何种权重条件下，有向树上的加权移位是共轭的、共轭共轭的、次正规的或完全超扩张的？
• RQ2 定义域包含关系 $\mathscr{D}(S_{\boldsymbol{\lambda}}) \subseteq \mathscr{D}(S_{\boldsymbol{\lambda}}^*)$ 或其逆关系的必要且充分条件是什么？
• RQ3 在树 ${\mathscr{T}}_{\eta,\kappa}$ 上的次正规性如何与经典单边加权移位的 $k$-步向后可延拓性相关？
• RQ4 对于有向树 ${\mathscr{T}}_{2,1}$ 上的加权移位，$p$-共轭性与参数空间 $\varDelta_p$ 之间的精确关系是什么？
• RQ5 能否在特定有向树上构造次正规和完全超扩张加权移位的模型？其权重需满足什么条件？

主要发现

• 在 ${\mathscr{T}}_{\eta,\kappa}$ 上的加权移位 $S_{\boldsymbol{\lambda}}$ 是 $p$-共轭的，当且仅当权重满足：对所有 $i,j$，有 $|\lambda_{i,j}| \leq |\lambda_{i,j+1}|$，$\left(\sum |\lambda_{i,1}|^2\right)^{p-1} \left(\sum \frac{|\lambda_{i,1}|^2}{|\lambda_{i,2}|^{2p}}\right) \leq 1$，且当 $\kappa \geq 1$ 时，有 $|\lambda_0|^2 \leq \sum |\lambda_{i,1}|^2$。
• 对于树 ${\mathscr{T}}_{2,1}$，$S_{\boldsymbol{\lambda}}$ 是 $p$-共轭的，当且仅当 $\lambda_0 \leq 1$ 且 $(a,b) \in \varDelta_p = \{(x,y) \in \mathbb{R}^2 : x,y > 0, x^{2p} + y^{2p} \leq 2\}$。
• 交集 $\varDelta_\infty = \bigcap_{p>0} \varDelta_p$ 等于 $\{(x,y) : 0 < x \leq 1, 0 < y \leq 1\}$，因此 $S_{\boldsymbol{\lambda}}$ 是 $\infty$-共轭的当且仅当 $\lambda_0, a, b \leq 1$。
• 在 ${\mathscr{T}}_{2,1}$ 上，$S_{\boldsymbol{\lambda}}$ 是次正规的，当且仅当 $a^2 + b^2 = 2$ 且 $\frac{a^4 + b^4}{2} \leq \frac{1}{\lambda_0^2}$。
• 若 $a^2 + b^2 = 2$ 且 $(a,b) \neq (1,1)$，则 $S_{\boldsymbol{\lambda}}$ 是次正规但不是 $\infty$-共轭的；而若 $(a,b) \in \varDelta_\infty \setminus \{(1,1)\}$，则它是 $\infty$-共轭的但不是次正规的。
• 算子 $S_{\boldsymbol{\lambda}}$ 是等距算子当且仅当 $\lambda_0 = a = b = 1$，此时也蕴含次正规性和 $\infty$-共轭性。