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 $(M_2(\mathsf {C}), \frac{1}{2}Tr)*(M_2(\mathsf{C}), \frac{1}{2}Tr)$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}Tr)*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}Tr)$, 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 $X eq λ 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}Tr)*(M_2(\mathsf{C}), \frac{1}{2}Tr)$.