Abstract: 
We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u_n+u_{\infty }$ where $u_n$ and $u_{\infty }$ are the generators of ${\sym Z}_n$ and ${\sym Z}$ respectively, in the free product ${\sym Z}_n\mathbin {*}{\sym Z}$, or elliptic elements of the form $S_{\alpha }+iS_{\beta }$ where $S_\alpha $ and $S_\beta $ are free semicircular elements of variance $\alpha $ and $\beta $.

1. CNRS
   DMA
   45 rue d'Ulm
   75005 Paris, France
   
2. Institut für Analysis und Numerik
   Johannes Kepler Universität Linz
   4040 Linz, Austria
   and
   Équipe d'Analyse
   Université Paris 6
   Boîte 186, Tour 46-0
   4, place Jussieu
   75252 Paris Cedex 05, France