Free Compressions of R-Diagonal Random Variables and the Semigroup of Brown Measures

Abstract We investigate the Brown measures of compressions of $$ R $$ R -diagonal random variables, extending previous results to include unbounded cases. For random variables with finite variance, we demonstrate that the Brown measures of their compressions converge to the uniform distribution on the unit disk. In the case of infinite variance, we characterize the Brown measures that remain stable under the compression operation and explore their properties in detail.