A moment-free R-transform for an unbounded R-diagonal pair

Definitions and mathematical statement

The source asks for the joint $R$-transform of $(c,c^*)$ when $c$ is unbounded and $R$-diagonal. Its appended "Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)" is the following section heading. The source specifies neither moment assumptions nor whether a formal cumulant series or an analytic transform is intended.

In a finite tracial von Neumann algebra, an affiliated $R$-diagonal distribution can be represented by $uh$, with $h\ge0$ affiliated and a Haar unitary $u$ free from $h$. Its moment-free distribution within this class is determined by the spectral probability law $\mu_h$ of $h=|c|$. Let $\nu$ be its symmetrization on ${\mathbb{R}}$. For $z$ in the upper half-plane define \[ G_\nu(z)=\int\frac{{\,\mathrm{d}}\nu(t)}{z-t},\qquad F_\nu(z)=1/G_\nu(z). \] The analytic Voiculescu transform uses the appropriate inverse branch $F_\nu^{-1}$ near infinity on a nontangential domain; equivalently an analytic $R$-transform is $G_\nu^{-1}(w)-1/w$ on the corresponding near-zero domain. No finite moments or analytic extension through zero is automatic. The aid reports that this transform is information-complete for the $R$-diagonal class and that symmetrized modulus laws linearize free sums: \[ \widetilde\mu_{|c+d|}=\widetilde\mu_{|c|}\boxplus\widetilde\mu_{|d|} \] for the appropriate free affiliated $R$-diagonal operators, with sum interpreted as an affiliated closure.

When moments through degree $2N$ exist, the aid identifies the available alternating determining coefficients with scalar even cumulants of $\nu$: \[ \alpha_k=\kappa_{2k}(\nu)\quad(1\le k\le N). \] If all moments are available, the tracial formal joint series is \[ R_{c,c^*}(z_1,z_2)=\sum_{k\ge1}\alpha_k \left[(z_1z_2)^k+(z_2z_1)^k\right], \] with nonalternating cumulants zero. Without the requisite moments this series is undefined, and even all finite moments do not by themselves give a convergent series at zero.

For a moment-free bounded regularization, the aid sets \[ b_s=c(1+s c^*c)^{-1/2},\qquad s>0,\qquad \|b_s\|\le s^{-1/2}. \] It is $R$-diagonal and has an ordinary joint $R$-transform. Its first alternating coefficient is \[ \alpha_1(s)=\tau(b_s^*b_s)=\int_0^\infty\frac{t^2}{1+st^2}\,{\,\mathrm{d}}\mu_h(t). \] The function of all $s>0$ determines $\mu_h$, since $1-s\alpha_1(s)=\int(1+st^2)^{-1}\,{\,\mathrm{d}}\mu_h(t)$ is a Stieltjes encoding of the squared-modulus law. The aid also states that every alternating cumulant justified by a finite $2k$-th modulus moment is recovered as $\lim_{s\downarrow0}\alpha_k(b_s)$. This supplies analytic and regularized alternatives to the possibly nonexistent literal joint cumulant transform; it does not silently impose moments on the original unbounded input.

Short English statement

An unbounded $R$-diagonal pair need not have the moments required for a formal joint $R$-series. The aid instead uses the analytic transform of the symmetrized modulus and bounded regularizations. The entire unbounded distribution is encoded by the first regularized coefficient as a function of its damping parameter.

Sources

• Local UnsolvedMath registry, permanent ID 20002691, source code catalogue entry; created and updated 14 August 2026. Canonical aim-probability-notes.json, zero-based notes[132], tag question, item 31.
• AIM "Free Analysis," physical/PDF page 7 as described by the supplied verification: https://aimath.org/WWN/freeanalysis/freeanalysis.pdf. The verification says it inspected the page visually; no such visit is claimed here. The record is marked exact and includes the following section heading. Original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. No complete authoritative bibliographic references or full analytic-domain hypotheses are supplied for the transform and regularization assertions.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.