Publicação
On essential norms of singular integral operators with constant coefficients and of the backward shift
| Resumo: | Let X be a rearrangement-invariant Banach function space on the unit circle T and let H[X] be the abstract Hardy space built upon X. We prove that if the Cauchy singular integral operator (Formula presented) is τ−t bounded on the space X, then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator aI + bH with a, b ∈ C, acting on the space X, coincide. We also show that similar equalities hold for the backward shift operator (Formula presented) on the abstract Hardy space H[X]. Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priložen. 9 (1975), pp. 73-74] for the operator aI + bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S. |
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| Autores principais: | Karlovych, Oleksiy |
| Outros Autores: | Shargorodsky, Eugene |
| Assunto: | abstract Hardy singular integral operator backward shift operator essential norm measure of noncompactness norm Rearrangement-invariant Banach function space Algebra and Number Theory Analysis Discrete Mathematics and Combinatorics Geometry and Topology |
| Ano: | 2022 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade Nova de Lisboa |
| Idioma: | inglês |
| Origem: | Repositório Institucional da UNL |
| Resumo: | Let X be a rearrangement-invariant Banach function space on the unit circle T and let H[X] be the abstract Hardy space built upon X. We prove that if the Cauchy singular integral operator (Formula presented) is τ−t bounded on the space X, then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator aI + bH with a, b ∈ C, acting on the space X, coincide. We also show that similar equalities hold for the backward shift operator (Formula presented) on the abstract Hardy space H[X]. Our results extend those by Krupnik and Polonskiĭ [Funkcional. Anal. i Priložen. 9 (1975), pp. 73-74] for the operator aI + bH and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator S. |
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