TY - JOUR

T1 - On essential norms of singular integral operators with constant coefficients and of the backward shift

AU - Karlovych, Oleksiy

AU - Shargorodsky, Eugene

N1 - Funding Information:
info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00297%2F2020/PT#
Received by the editors November 5, 2021. 2020 Mathematics Subject Classification. Primary 45E05, 46E30, 47B38. Key words and phrases. Rearrangement-invariant Banach function space, abstract Hardy singular integral operator, backward shift operator, norm, essential norm, measure of non-compactness.
Publisher Copyright:
© 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0).

PY - 2022/3/22

Y1 - 2022/3/22

N2 - 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.

AB - 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.

KW - abstract Hardy singular integral operator

KW - backward shift operator

KW - essential norm

KW - measure of noncompactness

KW - norm

KW - Rearrangement-invariant Banach function space

UR - http://www.scopus.com/inward/record.url?scp=85132267275&partnerID=8YFLogxK

U2 - 10.1090/bproc/118

DO - 10.1090/bproc/118

M3 - Article

AN - SCOPUS:85132267275

SN - 2330-1511

VL - 9

SP - 60

EP - 70

JO - Proceedings of the American Mathematical Society, Series B

JF - Proceedings of the American Mathematical Society, Series B

ER -