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Positivity conditions and standard models for commuting multioperators. (English) Zbl 0842.47005
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 347-365 (1995).
Summary: Positivity plays an important role in the Hilbert space operator theory. The simplest positivity condition is, perhaps, that characterizing a Hilbert space contraction $$C$$, i.e., $$1- C^* C\geq 0$$, where 1 denotes the identity of the space. The aim of this work is to present natural extensions of this positivity condition, valid for one or several (commuting) operators, and to derive some of their consequences. More precisely, we exhibit examples of operators that satisfy such conditions, which are called standard models. Then we partially describe the structure of arbitrary families of operators satisfying a given positivity condition, showing that such a family is essentially the restriction of the standard model to an invariant subspace, modulo an additional term of a related type. We obtain statements which extend and unify the corresponding ones from Curto, Müller and the author, using similar techniques.
For the entire collection see [Zbl 0819.00022].

##### MSC:
 47A13 Several-variable operator theory (spectral, Fredholm, etc.) 47A45 Canonical models for contractions and nonselfadjoint linear operators 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)