{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T09:37:28Z","timestamp":1773481048684,"version":"3.50.1"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T00:00:00Z","timestamp":1731024000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T00:00:00Z","timestamp":1731024000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eanncia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020, UIDP\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020, UIDP\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005855","name":"Universidade Nova de Lisboa","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005855","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bol. Soc. Mat. Mex."],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Let <jats:italic>X<\/jats:italic> be a Banach function space over the unit circle such that the Riesz projection <jats:italic>P<\/jats:italic> is bounded on <jats:italic>X<\/jats:italic> and let <jats:italic>H<\/jats:italic>[<jats:italic>X<\/jats:italic>] be the abstract Hardy space built upon <jats:italic>X<\/jats:italic>. We show that the essential norm of the Toeplitz operator <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$T(a):H[X]\\rightarrow H[X]$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>[<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>]<\/mml:mo>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>[<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>]<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> coincides with <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Vert a\\Vert _{L^\\infty }$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>\u2016<\/mml:mo>\n                      <mml:mi>a<\/mml:mi>\n                      <mml:mo>\u2016<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msup>\n                      <mml:mi>L<\/mml:mi>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> for every <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$a\\in C+H^\\infty $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> if and only if the essential norm of the backward shift operator <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$T(\\textbf{e}_{-1}):H[X]\\rightarrow H[X]$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>e<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mo>-<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:mrow>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>[<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>]<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>[<\/mml:mo>\n                      <mml:mi>X<\/mml:mi>\n                      <mml:mo>]<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is equal to one, where <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textbf{e}_{-1}(z)=z^{-1}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. This result extends an observation by B\u00f6ttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.<\/jats:p>","DOI":"10.1007\/s40590-024-00689-2","type":"journal-article","created":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T11:29:15Z","timestamp":1731065355000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces"],"prefix":"10.1007","volume":"31","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":[[2024,11,8]]},"reference":[{"key":"689_CR1","volume-title":"Interpolation of Operators. Pure and Applied Mathematics","author":"C Bennett","year":"1988","unstructured":"Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston (1988)"},{"issue":"4","key":"689_CR2","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1007\/BF01199302","volume":"11","author":"A B\u00f6ttcher","year":"1988","unstructured":"B\u00f6ttcher, A., Krupnik, N., Silbermann, B.: A general look at local principles with special emphasis on the norm computation aspect. Integral Equ. Oper. Theory 11(4), 455\u2013479 (1988)","journal-title":"Integral Equ. Oper. Theory"},{"key":"689_CR3","series-title":"Springer Monographs in Mathematics","volume-title":"Analysis of Toeplitz Operators","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":"689_CR4","volume-title":"Bounded Analytic Functions, Volume 236 of Graduate Texts in Mathematics","author":"JB Garnett","year":"2007","unstructured":"Garnett, J.B.: Bounded Analytic Functions, Volume 236 of Graduate Texts in Mathematics, 1st edn. Springer, New York (2007)","edition":"1"},{"key":"689_CR5","doi-asserted-by":"crossref","first-page":"347","DOI":"10.4064\/sm-31-4-347-362","volume":"31","author":"IC Gohberg","year":"1968","unstructured":"Gohberg, I.C., Krupnik, N.J.: The spectrum of singular integral operators in $$L_{p}$$ spaces. Studia Math. 31, 347\u2013362 (1968)","journal-title":"Studia Math."},{"issue":"2","key":"689_CR6","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1006\/jfan.2000.3616","volume":"175","author":"B Hollenbeck","year":"2000","unstructured":"Hollenbeck, B., Verbitsky, I.E.: Best constants for the Riesz projection. J. Funct. Anal. 175(2), 370\u2013392 (2000)","journal-title":"J. Funct. Anal."},{"issue":"2","key":"689_CR7","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s43036-021-00128-3","volume":"6","author":"A Karlovich","year":"2021","unstructured":"Karlovich, A.: Noncompactness of Toeplitz operators between abstract Hardy spaces. Adv. Oper. Theory 6(2), 29 (2021)","journal-title":"Adv. Oper. Theory"},{"key":"689_CR8","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1090\/bproc\/118","volume":"9","author":"O Karlovych","year":"2022","unstructured":"Karlovych, O., Shargorodsky, E.: On the essential norms of singular integral operators with constant coefficients and of the backward shift. Proc. Am. Math. Soc. Ser. B 9, 60\u201370 (2022)","journal-title":"Proc. Am. Math. Soc. Ser. B"},{"issue":"1","key":"689_CR9","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1007\/s00020-023-02725-8","volume":"95","author":"O Karlovych","year":"2023","unstructured":"Karlovych, O., Shargorodsky, E.: The Coburn lemma and the Hartman-Wintner-Simonenko theorem for Toeplitz operators on abstract Hardy spaces. Integral Equ. Oper. Theory 95(1), 6 (2023)","journal-title":"Integral Equ. Oper. Theory"},{"issue":"12","key":"689_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2023.110158","volume":"285","author":"O Karlovych","year":"2023","unstructured":"Karlovych, O., Shargorodsky, E.: When are the norms of the Riesz projection and the backward shift operator equal to one? J. Funct. Anal. 285(12), 110158 (2023)","journal-title":"J. Funct. Anal."},{"key":"689_CR11","volume-title":"Hardy Spaces, Volume 179 of Cambridge Studies in Advanced Mathematics","author":"N Nikolski","year":"2019","unstructured":"Nikolski, N.: Hardy Spaces, Volume 179 of Cambridge Studies in Advanced Mathematics, French Cambridge University Press, Cambridge (2019)","edition":"French"},{"issue":"2","key":"689_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2020.108835","volume":"280","author":"E Shargorodsky","year":"2021","unstructured":"Shargorodsky, E.: On the essential norms of Toeplitz operators with continuous symbols. J. Funct. Anal. 280(2), 108835 (2021)","journal-title":"J. Funct. Anal."}],"container-title":["Bolet\u00edn de la Sociedad Matem\u00e1tica Mexicana"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40590-024-00689-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40590-024-00689-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40590-024-00689-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T09:53:45Z","timestamp":1740995625000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40590-024-00689-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,8]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["689"],"URL":"https:\/\/doi.org\/10.1007\/s40590-024-00689-2","relation":{},"ISSN":["1405-213X","2296-4495"],"issn-type":[{"value":"1405-213X","type":"print"},{"value":"2296-4495","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,8]]},"assertion":[{"value":"22 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"8"}}