On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

2024/08/25 by Karlovych, Oleksiy, Shargorodsky, Eugene
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2408.13907

Abstract

Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→ H[X] coincides with ‖a‖_L^∞ for every a∈ C+H^∞ if and only if the essential norm of the backward shift operator T(e_-1):H[X]→ H[X] is equal to one, where e_-1(z)=z^-1. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

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