# 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 \be \label{q-contr} T_1T_1^* + \cdots + T_dT_d^* \le \1 . \eqno{\indent(0.1)}\ee 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.

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