COMPACT DIFFERENCE OF COMPOSITION OPERATORS ON THE HARDY SPACES

Boo Rim Choe; Koeun Choi; Hyung Woon Koo; Inyoung Park

doi:10.1090/btran/126

COMPACT DIFFERENCE OF COMPOSITION OPERATORS ON THE HARDY SPACES

Boo Rim Choe, Koeun Choi, Hyung Woon Koo, Inyoung Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of cer-tain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

Original language	English
Pages (from-to)	733-756
Number of pages	24
Journal	Transactions of the American Mathematical Society Series B
Volume	9
Issue number	24
DOIs	https://doi.org/10.1090/btran/126
Publication status	Published - 2022

Keywords

bounded multiplicity
compactness
Difference of composition operator
Hardy space

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.1090/btran/126

Cite this

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title = "COMPACT DIFFERENCE OF COMPOSITION OPERATORS ON THE HARDY SPACES",

abstract = "Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of cer-tain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.",

keywords = "bounded multiplicity, compactness, Difference of composition operator, Hardy space",

author = "Choe, {Boo Rim} and Koeun Choi and Koo, {Hyung Woon} and Inyoung Park",

note = "Funding Information: Received by the editors June 5, 2022. 2020 Mathematics Subject Classification. Primary 47B33; Secondary 30H10, 30H20. Key words and phrases. Difference of composition operator, compactness, Hardy space, bounded multiplicity. The first author was supported by NRF (2018R1D1A1B07041183) of Korea, the third author was supported by NRF (2022R1F1A1063305) of Korea and the fourth author was supported by NRF (2021R1I1A1A01047051) of Korea and NRF (2020R1A4A3079066) of the Ministry of Science and ICT of Korea. Publisher Copyright: {\textcopyright} 2022 by the author(s).",

year = "2022",

doi = "10.1090/btran/126",

language = "English",

volume = "9",

pages = "733--756",

journal = "Transactions of the American Mathematical Society Series B",

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number = "24",

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T1 - COMPACT DIFFERENCE OF COMPOSITION OPERATORS ON THE HARDY SPACES

AU - Choe, Boo Rim

AU - Choi, Koeun

AU - Koo, Hyung Woon

AU - Park, Inyoung

N1 - Funding Information: Received by the editors June 5, 2022. 2020 Mathematics Subject Classification. Primary 47B33; Secondary 30H10, 30H20. Key words and phrases. Difference of composition operator, compactness, Hardy space, bounded multiplicity. The first author was supported by NRF (2018R1D1A1B07041183) of Korea, the third author was supported by NRF (2022R1F1A1063305) of Korea and the fourth author was supported by NRF (2021R1I1A1A01047051) of Korea and NRF (2020R1A4A3079066) of the Ministry of Science and ICT of Korea. Publisher Copyright: © 2022 by the author(s).

PY - 2022

Y1 - 2022

N2 - Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of cer-tain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

AB - Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of cer-tain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

KW - bounded multiplicity

KW - compactness

KW - Difference of composition operator

KW - Hardy space

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JF - Transactions of the American Mathematical Society Series B

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ER -

COMPACT DIFFERENCE OF COMPOSITION OPERATORS ON THE HARDY SPACES

Abstract

Keywords

ASJC Scopus subject areas

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