On spectra and Brown’s spectral measures of elements in free products of matrix algebras - Fang, Junsheng; Hadwin, Don; Ma, Xiujuan (Mathematica Scandinavica, 2008)
We compute spectra and Brown measures of some non self-adjoint operators in (M_2(C), 12Tr)*(M_2(C), 12Tr), the reduced free product von Neumann algebra of M_2(C) with M_2(C). Examples include AB and A+B, where A and B are matrices in (M_2(C), 12Tr)*1 and 1*(M_2(C), 12Tr), 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 eq λ 1, then X has a nontrivial hyperinvariant subspace affiliated with (M_2(C), 12Tr)*(M_2(C), 12Tr).