CHARACTERIZATION PROBLEMS FOR LINEAR FORMS WITH

Abstract. Let T1,..., Tn denote free random variables. For two linear forms L1 =∑n j=1 ajTj and L2 = ∑n j=1 bjTj with real coefficients aj and bj we shall describe all distributions of T1,..., Tn such that L1 and L2 are free. For identically distributed free random variables T1,..., Tn with distribution µ we establish necessary and sufficient conditions on the coefficients aj, bj, j = 1,..., n, such that the statements: (i) µ is a centered semicircular distribution; and (ii) L1 and L2 are identically distributed (L1 D = L2); are equivalent. In the proof we give a complete characterization of all sequences of free cumulants of measures with compact support and with a finite number of non zero entries. The characterization is based on topological properties of regions defined by means of the Voiculescu transform φ of such sequences. 1