Converse of the Hilbert–Schmidt sufficient condition

Determine whether, for injective non-negative self-adjoint operators S and T satisfying S∼T, the operator L_{S,T}=\overline{S^{1/2}T^{-1/2}-1} must be Hilbert–Schmidt.

Background

The paper defines S∼T by requiring weak equivalence of S and T together with the Hilbert–Schmidt property of K_{S,T}=\overline{T{-1/2}S{1/2}}\,\overline{S{1/2}T{-1/2}-1}. Proposition \ref{def of L} proves that the stronger-looking condition that L_{S,T}=\overline{S{1/2}T{-1/2}-1} is Hilbert–Schmidt is sufficient for S∼T.

The unresolved issue is whether the converse implication holds in general. The paper subsequently proves that the converse does hold after replacing S and T by powers Sp and Tp for 0<p<1, but it does not resolve the original case p=1.

References

Is the converse of Proposition \ref{def of L} true? That is, if $S,T\in\mathcal{S!A}+(\mathscr{H})$ satisfy $S\sim T$, does it follow that $L{S,T}$ is Hilbert--Schmidt?

— Representation Theory of Canonical Commutation Relations Arising from the Quantization of the Klein-Gordon Equation with an External Potential  (2609.05237 - Matsuzawa et al., 4 Sep 2026) in Question labeled “rem of def of L,” Section 2, subsection “Equivalence relation”