Abstract
In this article 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.
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