# Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 197-212.

Powers of $R$-diagonal elements**Authors**: Flemming Larsen

**Author institution:**University of South Denmark, Odense University Campus, Campusvej 55, DK--5230 Odense M, Denmark

**Summary:**We prove that if $(a,b)$ is an $R$-diagonal pair in some non-commu\-tative probability space $(A,\phi)$ then $(a^p,b^p)$ is $R$-diagonal too and we compute the determining series $f_{(a^p,b^p)}$ in terms of the distribution of $ab$. We give estimates of the upper and lower bounds of the support of free multiplicative convolution of probability measures compactly supported on $[0,\infty[$, and use the results to give norm estimates of powers of $R$-diagonal elements in finite von~Neumann algebras. Finally we compute norms, distributions and $R$-transforms related to powers of the circular element.

**Keywords:**Free probability theory, $R$-diagonal elements, operator algebras

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