A property of Cauchy–Stieltjes kernel families based on dilation of measures

Abstract In this paper, we introduce a property of the inverse Semicircle and the free Gamma laws based on the dilation of measures in the context of Cauchy–Stieltjes Kernel (CSK) families. Assume that the CSK family produced by a non-degenerate probability measure λ on ℝ {\mathbb{R}} with support limited from above is ℱ + ⁢ ( λ ) = { ???? m λ ⁢ ( d ⁢ y ) : m ∈ ( m 1 λ , m + λ ) } {\mathcal{F}_{+}}(\lambda)=\{\mathcal{Q}_{m}^{\lambda}(dy):m\in(m_{1}^{\lambda% },m_{+}^{\lambda})\} . For α ∈ ℝ ∖ 0 {\alpha\in\mathbb{R}\setminus 0} , consider ???? α {\mathbf{H}_{\alpha}} , x ↦ α ⁢ x {x\mapsto\alpha x} and provide the set of measures ???? α ⁢ ( ℱ + ⁢ ( λ ) ) = { ???? α ⁢ ( ???? m λ ⁢ ( d ⁢ y ) ) : m ∈ ( m 1 λ , m + λ ) } . \mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))=\{\mathbf{H}_{\alpha}(\mathcal% {Q}_{m}^{\lambda}(dy)):m\in(m_{1}^{\lambda},m_{+}^{\lambda})\}. Let us say that α > 0 {\alpha>0} . We demonstrate that if ???? α ⁢ ( ℱ + ⁢ ( λ ) ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))} is a re-parametrization of ℱ + ⁢ ( λ ) {{\mathcal{F}_{+}}(\lambda)} (i.e., ???? α ⁢ ( ℱ + ⁢ ( λ ) ) = ℱ + ⁢ ( λ ) {\mathbf{H}_{\alpha}({\mathcal{F}_{+}}(\lambda))={\mathcal{F}_{+}}(\lambda)} ), then λ is either the free Gamma type law or the inverse Semicircle type law up to scaling.