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 P on the Lebesgue space L p lead to ‖P‖ L p →L p =1/sin⁡(π/p) for every p∈(1,∞). Hence L 2 is the only space among all Lebesgue spaces L p for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces L p . We prove that the norm of P is equal to one on the space X if and only if X coincides with L 2 and there exists a constant C∈(0,∞) such that ‖f‖ X =C‖f‖ L 2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.