Characterization Problems for Linear Forms with Free Summands

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.