The Matsaev (1971) conjecture is false for every p ≠ 2

The Matsaev (1971) conjecture asserts that for 1 < p < ∞, every contraction T on an Lp space and every polynomial P satisfy ‖P(T)‖ ≤ ‖P(S)‖, where S is the right shift on ℓp(ℕ); for p = 2 this is von Neumann's inequality. Drury (2011) disproved it for p = 4, and hence, by duality, for p = 4/3, with a computer-assisted example. We prove that it fails for every p ∈ (1, ∞) ∖ {2}, even with T a real 2×2 matrix and P a real quadratic polynomial. Let Pa(z) = 1 + az − Bz2 with B > 0. For p ≠ 2 and small a > 0 we show that ‖Pa(S)‖ = ‖Pa(J)‖, where J is the rotation of ℓp2 (ℂ2 with the ℓp norm) by a right angle. For p > 2 the bound ‖Pa(S)‖ ≤ ‖Pa(J)‖ comes from a Bellman function: a function G of two real variables such that, for every real sequence x ∈ ℓp(ℕ) (with x−1 = x−2 = 0) and every n ≥ 0, |(Pa(S)x)n|p ≤ ‖Pa(J)‖p|xn|p + G(xn−1, xn−2) − G(xn, xn−1); the G terms cancel in the sum over n. For 1 < p < 2 the bound follows by duality. If B > 1, a rotation by an angle slightly larger than π/2, scaled down to a contraction, gives a real 2×2 matrix T with ‖Pa(T)‖ > ‖Pa(J)‖ = ‖Pa(S)‖. In the other direction, the conjecture holds for every quadratic whose constant and leading coefficients have equal modulus. All results of this paper have also been proved in the Lean 4 proof assistant. An expanded version of this paper, with many other results, is posted on Zenodo: https://doi.org/10.5281/zenodo.23691816.