A simultaneous Wielandt positivity theorem

We consider matrix semigroups S which are closed under multiplication by complex scalars, and whose norm closure contains no zero-divisors. We show that when every non-zero S in S is indecomposable and the spectral radius of S is equal to the spectral radius of |S| for all S in S, then S is effectively positive, in the sense that there exists a diagonal unitary matrix D so that for each S in S, S = αS D|S|D −1 for some αS ∈ T. We also show the same conclusion holds even if individual indecomposability is weakened to indecomposability of the semigroup as a whole, as long as the semigroup is convex. We give examples showing that all hypotheses are required. We also extend some of these results to compact operators, under additional conditions.