Resolvents of R-Diagonal Operators

Abstract

We consider the resolvent $(λ-a)^{-1}$ of any $R$-diagonal operator $a$ in a $\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $R$-transform of the operator $|λ-c|^2$ where $c$ is Voiculescu's circular operator, and give an asymptotic formula for the negative moments of $|λ-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce {\em partition structure diagrams}, a new combinatorial structure arising in free probability.

29 pages, 12 figures, used gastex.sty