Scalar circular resolvent estimate and the sharp two-thirds obstruction

Definitions and mathematical statement

Let $(\mathcal M,\tau)$ be a finite tracial von Neumann algebra, and let $x,y$ be free unit-variance circular elements, free from the $\mathrm{II}_1$ algebra containing bounded $S,T$. For an affiliated measurable operator use $\|A\|_p=(\tau|A|^p)^{1/p}$, a quasi-norm when $0<p<1$. Inverses in the source are affiliated inverses, not necessarily bounded resolvents.

The source asks whether $x$ can replace $xy^{-1}$ in its Haagerup--Schultz inverse-difference estimate for $0<p<2/3$, with a constant depending only on $p$. Its extraction obscures the norm subscripts and a displayed reference label. Reading the two printed $p$'s as norm subscripts gives the target \[ \|(S+x)^{-1}-(T+x)^{-1}\|_p\le C_p\|S-T\|_p. \] If the original right side is the operator norm instead, that is a distinct operator-valued question requiring primary-source confirmation. The scalar partial results below use $S=sI,T=tI$, where both norms equal $|s-t|$, so they do not resolve that ambiguity. An exponent on $|s-t|$ is not silently substituted for the aid's Lipschitz reading.

The aid's scalar theorem is: for a unit-variance circular element $c$, every $0<q<1$ satisfies \[ \sup_{\lambda\in{\mathbb{C}}}\|(\lambda-c)^{-1}\|_q^q=M_q, \qquad M_q=\frac{2^{1-q}}\pi B\left(\frac{1-q}{2},\frac32\right), \] where $B(a,b)=\int_0^1t^{a-1}(1-t)^{b-1}\,{\,\mathrm{d}} t$ is the beta function. Equality in the supremum occurs at $\lambda=0$. Consequently, for every $0<p<1/2$ and $s,t\in{\mathbb{C}}$, \[ \|(s+c)^{-1}-(t+c)^{-1}\|_p \le M_{2p}^{1/p}|s-t|. \] The constant comes from the resolvent identity and noncommutative Hölder with two $L^{2p}$ inverse factors; it does not cover $p\ge1/2$, since the required inverse moment is then outside $q<1$.

The aid's sharp necessary threshold is separate. If a uniform scalar Lipschitz estimate at a given $p>0$ holds for all $s,t$, scalar difference quotients at zero and lower semicontinuity force $c^{-2}\in L^p$. The supplied Fuss--Catalan hard-edge calculation gives \[ c^{-2}\in L^p\quad\Longleftrightarrow\quad p<2/3. \] Here $c^{-2}=(c^2)^{-1}$ is not replaced by $|c|^{-2}$, since $c$ is nonnormal. Thus no such uniform scalar Lipschitz estimate is possible for $p\ge2/3$. This shows necessity of the source's upper endpoint, not sufficiency in the remaining strip.

The interval $1/2\le p<2/3$ even for scalar perturbations, and the general operator-valued $S,T$ assertion in the source's intended norm convention, remain unresolved in the supplied aid. The scalar result does not settle the proposed circular replacement of the original free Cauchy-type perturbation.

Short English statement

The aid proves the circular inverse-difference estimate for scalar perturbations when $p<1/2$, with an explicit inverse-moment constant. Scalar difference quotients exclude $p\ge2/3$. The intermediate scalar strip and the full operator-valued question remain open within the supplied result, with a damaged right-norm convention still requiring review.

Sources

• Local registry, permanent UnsolvedMath ID $20002677$, exact public code catalogue entry; canonical location aim-probability-notes.json, notes[118].
• AIM, Free Analysis, problem list dated 24 August 2006, section 0.6, "Invariant Subspaces for an Operator," question attributed to Haagerup; supplied reference https://aimath.org/WWN/freeanalysis/freeanalysis.pdf. The verification field reports prior visual checking of page 4 of the seven-page PDF to recover inverse placement and $2/3$. No PDF was accessed here.
• Haagerup--Schultz estimate, named in the source with an unresolved displayed label. The aid invokes a Fuss--Catalan hard-edge calculation but supplies no complete bibliographic references for either ingredient.
• Supplied research attempt 1: partial_result, run status unknown, rigor label proved, low novelty confidence; registry status partially_solved, classification PARTIAL-PROGRESS, created and updated 14 August 2026. The aid is machine generated, not peer reviewed, and requires independent verification before citation.