Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces. (28th September 2017)

• Record Type: Journal Article
• Authors: Karlovich, Alexei Yu.
• Other Names: Hudzik Henryk Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2017(2017)
• Publication Date: 2017-09-28
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2017/9768210
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : Let X be a Banach function space over the unit circle T and let H [ X ] be the abstract Hardy space built upon X . If the Riesz projection P is bounded on X and a ∈ L ∞, then the Toeplitz operator T a f = P ( a f ) is bounded on H [ X ] . We extend well-known results by Brown and Halmos for X = L 2 and show that, under certain assumptions on the space X, the Toeplitz operator T a is bounded (resp., compact) if and only if a ∈ L ∞ (resp., a = 0 ). Moreover, a L ∞ ≤ T a B ( H [ X ] ) ≤ P B ( X ) a L ∞ . These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.