We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial.
Agler, Jim; Knese, Greg; and McCarthy, John E., "Algebraic pairs of isometries" (2012). Mathematics Faculty Publications. 9.