Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 97-116.

A model theory for $q$-commuting contractive tuples

Authors: B.V. Rajarama Bhat (1), and Tirthankar Bhattacharyya (2)
Author institution: (1) Indian Statistical Institute, R.V. College Post, Bangalore 560059, India
(2) Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Summary: A contractive tuple is a tuple $(T_1, \ldots , T_d)$ of operators on a common Hilbert space such that
$\begin{equation} \label{q-contr} T_1T_1^* + \cdots + T_dT_d^* \le 1 . \end{equation}$
It is said to be $q$-commuting if $T_jT_i = q_{ij}T_iT_j$ for all $1 \le i < j \le d$, where $q_{ij}$, $1\leq i< j \leq d$ are complex numbers. These are higher-dimensional and non-commutative generalizations of a contraction. A particular example of this is the $q$-commuting shift. In this note, we investigate model theory for $q$-commuting contractive tuples using representations of the $q$-commuting shift.

Keywords: Contractions, model theory, dilation, complete positivity

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