Operators Cauchy dual to 2-hyperexpansive operators: the multivariable case
For \(p=(p_1,\dots,p_m)\in \mathbb N^m\), let \(|p|=\sum_{i=1}^m p_i\) and \(p\leq n\) if \(p_i\leq n_i\) for \(i=1,\dots,m\). For \(p\leq n\), let \({n \choose p}=\prod_{i=1}^m {n_i\choose p_i}\). Let \(\mathcal H \) be a Hilbert space, and \(\mathcal B(\mathcal H)\) the \(C^*\)-algebra of bounded linear operators on \(\mathcal H\). Let \(Q=(Q_1,\dots,Q_m)\) be a commuting \(m\)-tuple of positive, linear and bounded operators acting on \(\mathcal B(\mathcal H)\). The authors refer to \(Q\) as a \textit{generating m-tuple} on \(\mathcal H\). They are mainly interested in the following two cases of \(Q\) associated to a commuting \(m\)-tuple \(T=(T_1,\dots,T_m)\) with \(T_i \in \mathcal B(\mathcal H)\). \textit{Toral}: \[ Q_t=(Q_1,\dots,Q_m),\;\text{ where }Q_i(X)=T_i^*XT_i ~~(X \in \mathcal B(\mathcal H)). \] \textit{Spherical}: \[ Q_s=\sum_{i=1}^m T_i^* X T_i ~~(X \in \mathcal B(\mathcal H)). \] For \(Q\) a generating \(m\)-tuple and \(n \in \mathbb N^m\), set \[ B_n(Q)=\sum_{p\in\mathbb N^m,\;0\leq p \leq n} (-1)^{|p|}{ n\choose p} Q^p(I), \] where \(Q^0(I)=I\). It is said that \(Q\) is \textit{contractive} if \(B_n(Q)\geq 0\) for all \(|n|=1\), while \(Q\) is \textit{completely hypercontractive} if \(B_n(Q)\geq 0\) for all \(n \in \mathbb N^m\). Also, \(Q\) is \textit{p-hyperexpansive} if \(B_n(Q)\leq 0\) when \(0<|n|\leq p\), while it is \textit{completely hyperexpansive} if \(B_n(Q)\leq 0\) for all \(n\) with \(0<|n|.\) (See, for instance, [\textit{A. Athavale} and \textit{V. M. Sholapurkar}, Positivity 3, No. 3, 245--257 (1999; Zbl 0985.47003)]). The authors describe their work in their abstract as follows: ``As a natural outgrowth of the work done in [\textit{S. Chavan}, Proc. Edinb. Math. Soc., II. Ser. 50, No. 3, 637--652 (2007; Zbl 1155.47024); Stud. Math. 203, No. 2, 129--162 (2011; Zbl 1241.47019)], we introduce an abstract framework to study generating \(m\)-tuples, and use it to analyze hypercontractivity and hyperexpansivity in several variables. These two notions encompass (joint) hyponormality and subnormality, as well as toral and spherical isometric-ness; for instance, the Drury-Arvenson 2-shift is a spherical complete hyperexpansion. Our approach produces a unified theory that simultaneously covers toral and spherical hypercontractions (and hyperexpansions). As a byproduct, we arrive at a dilation theory for completely hypercontractive and completely hyperexpansive generating tuples. We can then analyze in detail the Cauchy duals of toral and spherical 2-hyperexpansive tuples. We also discuss various applications to the theory of hypercontractive and hyperexpansive tuples.''
scientific article; zbMATH DE number 6149102
scientific article