ABSTRACT

In this paper, we introduce the notions of the spherical p-hyponormality and the spherical log-hyponormality for commuting pairs of operators. We first prove that for a commuting pair $\mathbf{T}=(T_1,T_2)$ of operators, if $\mathbf{T}$ is jointly hyponormal, then $\mathbf{T}$ is spherically p-hyponormal for $0<p\le 1$. Second, we show that for $0<p,q\le 1$ and $q\le p$, a spherically p-hyponormal operator is spherically q-hyponormal. Third, we completely characterize the spherically p-hyponormal 2-variable weighted shifts. Fourth, we show that if $\mathbf{T}$ is Taylor invertible and spherically p-hyponormal, then $\mathbf{T}$ is spherically log-hyponormal. Moreover, it is possible to create a striking example regarding spherically log-hyponormal but not p-hyponormal. Furthermore, we show that the spherical log-hyponormality of $\mathbf{T}$ with a norm condition implies the spherical log-hyponormality for $\hat{\mathbf{T}}$, where $\hat{\mathbf{T}}$ is the spherical Aluthge transform of $\mathbf{T}$. Finally, in the case of 2-variable weighted shifts, we show that the spherical Aluthge transform does not preserve joint hyponormality, in sharp contrast with the 1-variable case.

摘要

在本文中，我们介绍了用于交换算子对的球面p超正态性和球面对数超正态性的概念。我们首先证明对于通勤对$\mathbf{吨}=({吨}_1,{吨}_2)$的运算符，如果$\mathbf{吨}$是联合超正态，那么$\mathbf{吨}$是球面p - 亚正态$0<p\le 1$. 其次，我们证明对于$0<p,q\le 1$和$q\le p$, 球面p次正态算子是球面q次正态算子。第三，我们完全描述了球形p次正态 2 变量加权偏移。第四，我们证明如果$\mathbf{吨}$是泰勒可逆且球面p超正态，那么$\mathbf{吨}$是球面对数次正态的。此外，可以创建一个关于球形对数次正态但不是p次正态的显着示例。此外，我们证明了$\mathbf{吨}$具有范数条件意味着球形对数次正态性$\widehat{\mathbf{吨}}$， 在哪里$\widehat{\mathbf{吨}}$是球面 Aluthge 变换$\mathbf{吨}$. 最后，在 2 变量加权偏移的情况下，我们表明球面 Aluthge 变换不保留关节超正态性，这与 1 变量情况形成鲜明对比。