Joint extensions in families of contractive commuting operator tuples
The elementary operation of restricting a linear transformation to an invariant subspace can be reversed only in particular situations, by a more complex and intricate process. The classical and best understood example is the restriction of a normal operator to an invariant subspace, giving rise by definition to a \textit{subnormal} operator. The theory of subnormal operators is intimately related to function theory of complex variable [see \textit{J.\,B.\thinspace Conway}, ``The theory of subnormal operators'' (Providence, RI:\ American Mathematical Society) (1991; Zbl 0743.47012)]. A very successful and lately widely amplified attempt to lay an abstract framework for operator extensions was proposed by \textit{J.\,Agler} in his article [``An abstract approach to model theory'', Pitman Res.\ Notes Math.\ Ser.\ 192, 1--23 (1989; Zbl 0788.47005)]. Specifically, Agler has introduced the notion of a family of tuples of Hilbert space operators by imposing the following axioms: uniform boundedness, invariance under restriction to joint invariant subspaces, invariance under countable direct sums and invariance under *-homomorphisms. The description of extremal elements in such a family of tuples of operators is paramount for obtaining functional models, that is, representations of abstract operators by concrete operations in a function space (such as multipliers or integral operators). For further developments, see \textit{J.\,Agler} and \textit{J.\,McCarthy} [``Pick interpolation and Hilbert function spaces'' (Providence, RI:\ American Mathematical Society) (2002; Zbl 1010.47001)]. The article under review treats from Agler's theory perspective the classes of spherical isometries, and dually, of row contractions. A detailed analysis of extremal elements and low rank extensions adds valuable new insights into both families of tuples of operators. As expected, a few preferred functional models dealing with analytic functions in the unit ball of \({\mathbb C}^n\) appear in the proofs.
scientific article; zbMATH DE number 5708608
scientific article