Numerical invariants for commuting isometric pairs Weiyong HeYueshi QinRongwei Yang 47A13commuting isometric pairfringe operatordefect operatorcongruence relation The fringe operator and the defect operator are defined for commuting isometric pairs $(V_1,V_2)$. Through an analysis on their relationship, as well as the structure of the defect operator, some numerical invariants for the commuting isometric pair $(V_1,V_2)$ are obtained. A new criterion is given to classify commuting isometric pairs, namely, the congruence relation. It turns out that, in many interesting cases, the congruence relation can be determined by a pair of nonnegative integers. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5409 10.1512/iumj.2015.64.5409 en Indiana Univ. Math. J. 64 (2015) 1 - 19 state-of-the-art mathematics http://iumj.org/access/