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=α _S D |S| D^-1 for some α _S ∈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.