On spectra and Brown's spectral measures of elements in free products of matrix algebras

We compute spectra and Brown measures of some non self-adjoint operators in (M2(C), 1 2 Tr) . (M2(C), 1 2 Tr), the reduced free product von Neumann algebra of M2(C) with M2(C). Examples include AB and A+B, where A and B are matrices in (M2(C), 1 2 Tr) . 1 and 1 . (M2(C), 1 2 Tr), respectively.We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if Tr(A) = Tr(B) = 0. We show that if X = AB or X = A + B and A,B are not scalar matrices, then the Brown measure of X is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if X = AB or X = A + B and X = ¥e1, then X has a nontrivial hyperinvariant subspace affiliated with (M2(C), 1 2 Tr) . (M2(C), 1 2 Tr).