# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 135-151.

The cyclic group and the transpose of an $R$-cyclic matrix

Authors:  Octavio Arizmendi (1), James A. Mingo (2)
Author institution: (1) Centro de Investigacion en Matematicas, Guanajuato, Mexic
(2) Department of Mathematics and Statistics, Queen's University, Jeffery Hall, Kingston, Ontario, K7L 3N6, Canada

Summary: We show that using the cyclic group the transpose of an $R$-cyclic matrix can be decomposed along diagonal parts into a sum of parts which are freely independent over diagonal scalar matrices. Moreover, if the $R$-cyclic matrix is self-adjoint then the off-diagonal parts are $R$-diagonal.

DOI: http://dx.doi.org/10.7900/jot.2019oct09.2281
Keywords: free probability, $R$-diagonal operators, $R$-cyclic operators