When is the Gohberg–Krupnik lower bound for aI+bH sharp? An expanded version, with other miscellanea

Let P+ and P− be the Riesz projections on the complex space Lp(T), 1 < p < ∞, onto the nonnegative and the negative frequencies, and let H = −i(P+ − P−), so that aI + bH = αP+ + βP− with α = a − ib and β = a + ib. Gohberg and Krupnik gave a lower bound for ‖aI + bH‖p, and Hollenbeck, Kalton and Verbitsky (2003) remarked that it is not sharp in general. We characterize when it is sharp. For p ≠ 2 and α ≠ 0 the bound is not sharp whenever the ratio β/α is not real, and it is sharp whenever β/α is real and nonpositive. There is an explicitly defined p0 ≈ 1.2336026 such that the bound is sharp for every real ratio exactly when p0 ≤ p ≤ p0/(p0 − 1); outside this interval it is not sharp for any real ratio below and sufficiently close to 1. For 1 < p < 2 and each real ratio λ = β/α ∈ (0, 1) we give an exact criterion: an inequality on the positive real axis, together with the condition that λ be smaller than an explicit threshold depending on p; the symmetries α ↔ β and p ↔ p′, where p′ = p/(p − 1), cover the remaining real ratios and exponents. For 1 < p < p0 the bound is sharp for all λ near 0, and on a band of intermediate ratios the outcome of the criterion is known only in part. The cases of H itself, of a single projection and of p = 2 are classical. We do not determine the norm where the bound is not sharp. Section 6 gives further results, including a wide-chord inequality that yields explicit lower bounds for the norm, strictly above the Gohberg–Krupnik bound, at two sample pairs (λ, p). The proof of the classification is computer-assisted and uses rigorous interval arithmetic on compact domains. The classification has also been proved in the Lean proof assistant, with the interval computations and two cited theorems (the Hollenbeck–Verbitsky norm of P+ and Cole's theorem) taken as given.