An Approximate, Multivariable Version of Specht’s Theorem

In this paper we provide generalizations of Specht’s Theorem which states that two n× n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A∗) and (B, B∗) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A∗) and (B, B∗) to coincide, but only to be close.