Computation of some examples of Brown's spectral measure in free probability
Tom 90 / 2001
Colloquium Mathematicum 90 (2001), 181-211
MSC: Primary 22D25, 46L54; Secondary 15A52, 43A05, 60J15.
DOI: 10.4064/cm90-2-3
Streszczenie
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 ${\mathbb Z}_n$ and ${\mathbb Z}$ respectively, in the free product ${\mathbb Z}_n\mathbin {*}{\mathbb Z}$, 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 $.
