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

Abstract

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 Lp lead to ‖P‖Lp→Lp=1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 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.