Characterization problems for linear forms with free summands

Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1=\sum_{j=1}^n a_jT_j$ and $L_2=\sum_{j=1}^n b_jT_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $μ$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,\,j=1,...,n,$ such that the statements:\quad $(i)$ $μ$ is a centered semicircular distribution; and $(ii)$ \, $L_1$ and $L_2$ are identically distributed ($L_1\stackrel{D}{=}L_2$); 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.

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