Remark: What the mass can and cannot cure

For $m=0,\beta =0$, Friedrichs is the only nonnegative extension in this ledger. For $m>0$, a logarithmic extension is nonnegative precisely when $m\ge {\kappa }_L$. The same mass-shift qualification applies to every $\lambda <0$ power-law mode. It would be incorrect to label all such massive operators “not nonnegative” merely because their transverse extension has a negative eigenvalue.