{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T09:37:24Z","timestamp":1773481044607,"version":"3.50.1"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T00:00:00Z","timestamp":1674000000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T00:00:00Z","timestamp":1674000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Integr. Equ. Oper. Theory"],"published-print":{"date-parts":[[2023,3]]},"abstract":"Abstract<\/jats:title>Let X<\/jats:italic> be a Banach function space on the unit circle $${\\mathbb {T}}$$<\/jats:tex-math>\n T<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, let $$X'$$<\/jats:tex-math>\n \n X<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be its associate space, and let H<\/jats:italic>[X<\/jats:italic>] and $$H[X']$$<\/jats:tex-math>\n \n H<\/mml:mi>\n [<\/mml:mo>\n \n X<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be the abstract Hardy spaces built upon X<\/jats:italic> and $$X'$$<\/jats:tex-math>\n \n X<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively. Suppose that the Riesz projection P<\/jats:italic> is bounded on X<\/jats:italic> and $$a\\in L^\\infty {\\setminus }\\{0\\}$$<\/jats:tex-math>\n \n a<\/mml:mi>\n \u2208<\/mml:mo>\n \n L<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n \\<\/mml:mo>\n \n {<\/mml:mo>\n 0<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that P<\/jats:italic> is bounded on $$X'$$<\/jats:tex-math>\n \n X<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. So, we can consider the Toeplitz operators $$T(a)f=P(af)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n f<\/mml:mi>\n =<\/mml:mo>\n P<\/mml:mi>\n (<\/mml:mo>\n a<\/mml:mi>\n f<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$T({\\overline{a}})g=P({\\overline{a}}g)$$<\/jats:tex-math>\n \n T<\/mml:mi>\n \n (<\/mml:mo>\n \n a<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n )<\/mml:mo>\n <\/mml:mrow>\n g<\/mml:mi>\n =<\/mml:mo>\n P<\/mml:mi>\n \n (<\/mml:mo>\n \n a<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n g<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on H<\/jats:italic>[X<\/jats:italic>] and $$H[X']$$<\/jats:tex-math>\n \n H<\/mml:mi>\n [<\/mml:mo>\n \n X<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively. In our previous paper, we have shown that if X<\/jats:italic> is not separable, then one cannot rephrase Coburn\u2019s lemma as in the case of classical Hardy spaces $$H^p$$<\/jats:tex-math>\n \n H<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$1<p<\\infty $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n <<\/mml:mo>\n p<\/mml:mi>\n <<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and guarantee that T<\/jats:italic>(a<\/jats:italic>) has a trivial kernel or a dense range on H<\/jats:italic>[X<\/jats:italic>]. The first main result of the present paper is the following extension of Coburn\u2019s lemma: the kernel of T<\/jats:italic>(a<\/jats:italic>) or the kernel of $$T({\\overline{a}})$$<\/jats:tex-math>\n \n T<\/mml:mi>\n (<\/mml:mo>\n \n a<\/mml:mi>\n \u00af<\/mml:mo>\n <\/mml:mover>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is trivial. The second main result is a generalisation of the Hartman\u2013Wintner\u2013Simonenko theorem saying that if T<\/jats:italic>(a<\/jats:italic>) is normally solvable on the space H<\/jats:italic>[X<\/jats:italic>], then $$1\/a\\in L^\\infty $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n a<\/mml:mi>\n \u2208<\/mml:mo>\n \n L<\/mml:mi>\n \u221e<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00020-023-02725-8","type":"journal-article","created":{"date-parts":[[2023,1,18]],"date-time":"2023-01-18T19:11:29Z","timestamp":1674069089000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Coburn Lemma and the Hartman\u2013Wintner\u2013Simonenko Theorem for Toeplitz Operators on Abstract Hardy Spaces"],"prefix":"10.1007","volume":"95","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-0561","authenticated-orcid":false,"given":"Oleksiy","family":"Karlovych","sequence":"first","affiliation":[]},{"given":"Eugene","family":"Shargorodsky","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,18]]},"reference":[{"key":"2725_CR1","unstructured":"Bennett, C., Sharpley, R.: Interpolation of Operators. Volume 129 of Pure and Applied Mathematics. Academic Press, Inc., Boston (1988)"},{"key":"2725_CR2","doi-asserted-by":"crossref","unstructured":"B\u00f6ttcher, A., Karlovich, Y.I.: Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Volume 154 of Progress in Mathematics. Birkh\u00e4user, Basel (1997)","DOI":"10.1007\/978-3-0348-8922-3"},{"key":"2725_CR3","volume-title":"Analysis of Toeplitz Operators. Springer Monographs in Mathematics","author":"A B\u00f6ttcher","year":"2006","unstructured":"B\u00f6ttcher, A., Silbermann, B.: Analysis of Toeplitz Operators. Springer Monographs in Mathematics, 2nd edn. Springer, Berlin (2006)","edition":"2"},{"key":"2725_CR4","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1307\/mmj\/1031732778","volume":"13","author":"LA Coburn","year":"1966","unstructured":"Coburn, L.A.: Weyl\u2019s theorem for nonnormal operators. Mich. Math. J. 13, 285\u2013288 (1966)","journal-title":"Mich. Math. J."},{"key":"2725_CR5","doi-asserted-by":"crossref","unstructured":"Douglas, R.G.: Banach Algebra Techniques in Operator Theory. Volume 179 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1998)","DOI":"10.1007\/978-1-4612-1656-8"},{"key":"2725_CR6","unstructured":"Duren, P.L.: Theory of $$H^{p}$$ Spaces. Volume 38 of Pure and Applied Mathematics. Academic Press, New York (1970)"},{"key":"2725_CR7","unstructured":"Garnett, J.B.: Bounded Analytic Functions. Volume 236 of Graduate Texts in Mathematics, 1st edn. Springer, New York (2007)"},{"key":"2725_CR8","volume-title":"One-Dimensional Linear Singular Integral Equations. I Introduction, Volume 53 of Operator Theory: Advances and Applications","author":"I Gohberg","year":"1992","unstructured":"Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations. I Introduction, Volume 53 of Operator Theory: Advances and Applications. Birkh\u00e4user, Basel (1992)"},{"key":"2725_CR9","volume-title":"Unbounded Linear Operators: Theory and Applications","author":"S Goldberg","year":"1966","unstructured":"Goldberg, S.: Unbounded Linear Operators: Theory and Applications. McGraw-Hill Book Co., New York (1966)"},{"key":"2725_CR10","doi-asserted-by":"crossref","unstructured":"Grafakos, L.: Classical Fourier Analysis. Volume 249 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2014)","DOI":"10.1007\/978-1-4939-1194-3"},{"key":"2725_CR11","doi-asserted-by":"publisher","first-page":"867","DOI":"10.2307\/2372661","volume":"76","author":"P Hartman","year":"1954","unstructured":"Hartman, P., Wintner, A.: The spectra of Toeplitz\u2019s matrices. Am. J. Math. 76, 867\u2013882 (1954)","journal-title":"Am. J. Math."},{"key":"2725_CR12","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1002\/mana.19770800111","volume":"80","author":"D Heunemann","year":"1977","unstructured":"Heunemann, D.: \u00dcber die normale Aufl\u00f6sbarkeit singul\u00e4rer Integraloperatoren mit unstetigem Symbol. Math. Nachr. 80, 157\u2013163 (1977)","journal-title":"Math. Nachr."},{"issue":"4","key":"2725_CR13","doi-asserted-by":"publisher","first-page":"436","DOI":"10.1007\/BF01194990","volume":"32","author":"A Karlovich","year":"1998","unstructured":"Karlovich, A.: Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces. Integral Equ. Oper. Theory 32(4), 436\u2013481 (1998)","journal-title":"Integral Equ. Oper. Theory"},{"issue":"3","key":"2725_CR14","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1216\/jiea\/1181074970","volume":"15","author":"A Karlovich","year":"2003","unstructured":"Karlovich, A.: Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. J. Integral Equ. Appl. 15(3), 263\u2013320 (2003)","journal-title":"J. Integral Equ. Appl."},{"key":"2725_CR15","doi-asserted-by":"crossref","unstructured":"Karlovich, A.: The Coburn-Simonenko theorem for Toeplitz operators acting between Hardy type subspaces of different Banach function spaces. Mediterr. J. Math. 15(3):Paper No. 91, 15 (2018)","DOI":"10.1007\/s00009-018-1139-3"},{"key":"2725_CR16","doi-asserted-by":"crossref","unstructured":"Karlovich, A.: Noncompactness of Toeplitz operators between abstract Hardy spaces. Adv. Oper. Theory 6(2):Paper No. 29, 10 (2021)","DOI":"10.1007\/s43036-021-00128-3"},{"issue":"1","key":"2725_CR17","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.jmaa.2018.11.022","volume":"472","author":"A Karlovich","year":"2019","unstructured":"Karlovich, A., Shargorodsky, E.: The Brown-Halmos theorem for a pair of abstract Hardy spaces. J. Math. Anal. Appl. 472(1), 246\u2013265 (2019)","journal-title":"J. Math. Anal. Appl."},{"issue":"4","key":"2725_CR18","doi-asserted-by":"publisher","first-page":"901","DOI":"10.1112\/plms.12206","volume":"118","author":"A Karlovich","year":"2019","unstructured":"Karlovich, A., Shargorodsky, E.: When does the norm of a Fourier multiplier dominate its $$L^\\infty $$ norm? Proc. Lond. Math. Soc. (3) 118(4), 901\u2013941 (2019)","journal-title":"Proc. Lond. Math. Soc. (3)"},{"key":"2725_CR19","doi-asserted-by":"crossref","unstructured":"Karlovych, O., Shargorodsky, E.: Toeplitz operators with non-trivial kernels and non-dense ranges on weak Hardy spaces. In: Toeplitz Operators and Random Matrices, Volume 289 of Operator Theory: Advances and Applications. Springer, Cham, 463\u2013476 (2022)","DOI":"10.1007\/978-3-031-13851-5_20"},{"key":"2725_CR20","unstructured":"Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)"},{"issue":"vyp. 1 (15)","key":"2725_CR21","first-page":"152","volume":"5","author":"J La\u012dterer","year":"1970","unstructured":"La\u012dterer, J.: The normal solvability of singular integral equations. Mat. Issled. 5(vyp. 1 (15)), 152\u2013159 (1970)","journal-title":"Mat. Issled."},{"issue":"1(23)","key":"2725_CR22","first-page":"72","volume":"7","author":"J La\u012dterer","year":"1972","unstructured":"La\u012dterer, J., Markus, A.: The normal solvability of singular integral operators in symmetic spaces. Mat. Issled. 7(1(23)), 72\u201382 (1972)","journal-title":"Mat. Issled."},{"key":"2725_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61631-0","volume-title":"Singular Integral Operators","author":"SG Mikhlin","year":"1986","unstructured":"Mikhlin, S.G., Pr\u00f6ssdorf, S.: Singular Integral Operators. Springer, Berlin (1986)"},{"key":"2725_CR24","unstructured":"Nikolski, N.: Hardy Spaces, Volume 179 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2019)"},{"key":"2725_CR25","doi-asserted-by":"crossref","unstructured":"Schechter, M.: Principles of Functional Analysis. Volume 36 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2002)","DOI":"10.1090\/gsm\/036"},{"key":"2725_CR26","doi-asserted-by":"publisher","first-page":"1091","DOI":"10.1070\/IM1968v002n05ABEH000706","volume":"2","author":"IB Simonenko","year":"1968","unstructured":"Simonenko, I.B.: Some general questions in the theory of the Riemann boundary problem. Math. USSR Izv. 2, 1091\u20131099 (1968)","journal-title":"Math. USSR Izv."}],"container-title":["Integral Equations and Operator Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00020-023-02725-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00020-023-02725-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00020-023-02725-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T09:06:57Z","timestamp":1678698417000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00020-023-02725-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,18]]},"references-count":26,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["2725"],"URL":"https:\/\/doi.org\/10.1007\/s00020-023-02725-8","relation":{},"ISSN":["0378-620X","1420-8989"],"issn-type":[{"value":"0378-620X","type":"print"},{"value":"1420-8989","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,18]]},"assertion":[{"value":"11 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"6"}}