Optimality of the quantitative stability exponent and constant

Determine whether the exponent 1/8 and the constant 6 in the quantitative estimate \({Im\,T}\leq 6{T}^{7/8}\varepsilon^{1/8}\) are optimal for bounded operators satisfying the stated approximate Fong–Tsui inequality.

Background

The paper derives a quantitative stability estimate showing that, when ∣T∣≤∣Re T∣+εI|T|\leq |Re\,T|+\varepsilon I and 0≤ε≤∥T∥0\leq\varepsilon\leq\|T\|, the imaginary part satisfies ∥Im T∥≤6∥T∥7/8ε1/8\|Im\,T\|\leq 6\|T\|^{7/8}\varepsilon^{1/8}. It proves only that exponents greater than $1/2$ cannot hold uniformly, leaving unresolved whether the exponent $1/8$, and the accompanying constant 6, can be improved.

References

This is an obstruction to exponents greater than 1/2, not a proof that 1/2 is attainable. Neither the exponent 1/8 nor the constant 6 in eq:holder is asserted to be optimal.

eq:holder:

B≤6N7/8ε1/8.{B}\leq6N^{7/8}\varepsilon^{1/8}.

— A General Proof of the Fong-Tsui Conjecture  (2609.10797 - Ma, 9 Sep 2026) in Section 5, Section “Sharpness restrictions and a hypothesis check”