Characterize operator pairs associated with the trace formula

Characterize the general properties of pairs of operators (T_+,T_-) whose resolvents are modeled by the difference-quotient operators R_\alpha on the reproducing-kernel spaces \mathfrak H(F) and \mathfrak H(F^{-1}), and determine when the trace formula established for such pairs applies.

Background

For t=-1, the paper constructs operators M_+ and M_- associated with a function F\in\mathcal{HP}{T,-1} and its inverse, and derives a trace formula for the difference of their resolvents after conjugation by multiplication operators. The subsequent remark abstracts this construction to a pair (T+,T_-) on a Hilbert space, represented through the same difference-quotient operators and multiplication by F{-1}. The authors explicitly state that the general properties of such operator pairs are unknown, leaving open a characterization of the pairs to which the trace formula can be applied.

References

We do not know what are the general properties of such a pair $(T_+,T_-)$, to which would be applicable the above trace formula.

— Hyperpositive functions, sector bounded functions and a trace formula  (2609.03403 - Alpay et al., 3 Sep 2026) in Section 5, subsection “Reproducing kernel Hilbert spaces and operator models,” final remark following the proof of the trace formula theorem