Sharp Shadowability Bounds for Normal Hyperbolic Linear Operators

Let \( A \) be an invertible normal hyperbolic operator on a complex Hilbert space \( H \). We obtain an exact spectral formula for the optimal uniform shadowability constant \( Shad(A) \), sharpening the general additive Green-operator estimates to a Euclidean combination of the stable and unstable spectral contributions. If both spectral components are nonempty, set \( \alpha(A)=1-\max\{|\lambda|:\lambda\in\sigma(A),\ |\lambda|<1\}, \) and \( \beta(A)=\min\{|\lambda|:\lambda\in\sigma(A),\ |\lambda|>1\}-1. \) We prove the exact formula \( Shad(A)=\left(\alpha(A)^{-2}+\beta(A)^{-2}\right)^{1/2}, \) with the corresponding one-sided formulas when only one spectral component is present. We further relate this exact constant to the robustness of hyperbolicity. Let \( r_{\mathrm{hyp}}(A) \) denote the operator-norm distance from \( A \) to the loss of hyperbolicity. We obtain the sharp universal ratio \( 1\le r_{\mathrm{hyp}}(A)Shad(A)\le\sqrt{2}. \) Both constants are optimal. In the mixed case, \( r_{\mathrm{hyp}}(A)Shad(A)=\left[1+\left(\frac{\min\{\alpha(A),\beta(A)\}}{\max\{\alpha(A),\beta(A)\}}\right)^2 \right]^{1/2}. \) Thus the normalized shadowability constant is determined exactly by the relative balance of the stable and unstable spectral gaps.