Operators Cauchy Dual to 2-Hyperexpansive Operators: The Multivariable Case

As a natural outgrowth of the work done in Chavan (Proc Edin Math Soc 50:637–652, 2007; Studia Math 203:129–162, 2011), 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–Arveson 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.