When are the norms of the Riesz projection and the backward shift operator equal to one?

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection Pon the Lebesgue space L-p lead to parallel to P parallel to(Lp -> Lp)= 1/ sin(pi/p) for every p is an element of(1, infinity). Hence L-2 is the only space among all Lebesgue spaces L-p for which the norm of the Riesz projection Pis equal to one. Banach function spaces Xare far-reaching generalisations of Lebesgue spaces L-p. We prove that the norm of Pis equal to one on the space Xif and only if Xcoincides with L-2 and there exists a constant C is an element of(0, infinity) such that parallel to f parallel to(X) = C parallel to f parallel to(L2) for all functions f is an element of X. Independently from this, we also show that the norm of Pon Xis equal to one if and only if the norm of the backward shift operator Son the abstract Hardy space H[X] built upon X is equal to one. (c) 2023 The Author(s). Published by Elsevier Inc.