Spectral results for free random variables

Abstract Let ( A , t r ) $\left(\mathcal{A},\mathrm{t}\mathrm{r}\right)$ be a von Neumann algebra with a faithful, normal trace t r : A → C $\mathrm{t}\mathrm{r} : \mathcal{A}\to \mathbb{C}$ . For each a ∈ A $a\in \mathcal{A}$ , define S ( λ , ε ) = t r [ log ( ( a − λ ) * ( a − λ ) + ε ) ] , λ ∈ C , ε > 0 , $$S\left(\lambda ,\varepsilon \right)=\mathrm{t}\mathrm{r}\left[\mathrm{log}\left({\left(a-\lambda \right)}^{{\ast}}\left(a-\lambda \right)+\varepsilon \right)\right],\quad \lambda \in \mathbb{C}, \varepsilon { >}0,$$ so that the limit as ɛ → 0 + of S is the log potential of the Brown measure of a . Suppose that for a fixed λ ∈ C $\lambda \in \mathbb{C}$ , the function ε ↦ ∂ S ∂ ε ( λ , ε ) = t r [ log (