{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T21:24:41Z","timestamp":1740173081627,"version":"3.37.3"},"reference-count":26,"publisher":"Wiley","license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/00297\/2013"],"award-info":[{"award-number":["UID\/MAT\/00297\/2013"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Function Spaces"],"published-print":{"date-parts":[[2017]]},"abstract":"<jats:p>Let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math> be a Banach function space over the unit circle <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi mathvariant=\"double-struck\">T<\/mml:mi><\/mml:mrow><\/mml:math> and let <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math> be the abstract Hardy space built upon <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>. If the Riesz projection <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><\/mml:math> is bounded on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>a<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math>, then the Toeplitz operator <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:msub><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mi>f<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>P<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>a<\/mml:mi><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> is bounded on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>. We extend well-known results by Brown and Halmos for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>X<\/mml:mi><mml:mo>=<\/mml:mo><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> and show that, under certain assumptions on the space <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>, the Toeplitz operator <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:msub><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> is bounded (resp., compact) if and only if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>a<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math> (resp., <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>a<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">0<\/mml:mn><\/mml:math>). Moreover, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:msub><mml:mrow><mml:mfenced open=\"\u2016\" close=\"\u2016\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:msub><mml:mo>\u2264<\/mml:mo><mml:msub><mml:mrow><mml:mfenced open=\"\u2016\" close=\"\u2016\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">B<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>H<\/mml:mi><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msub><mml:mo>\u2264<\/mml:mo><mml:msub><mml:mrow><mml:mfenced open=\"\u2016\" close=\"\u2016\" separators=\"|\"><mml:mrow><mml:mi>P<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">B<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>X<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mfenced open=\"\u2016\" close=\"\u2016\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:msub><\/mml:math>. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.<\/jats:p>","DOI":"10.1155\/2017\/9768210","type":"journal-article","created":{"date-parts":[[2017,9,28]],"date-time":"2017-09-28T17:07:17Z","timestamp":1506618437000},"page":"1-8","source":"Crossref","is-referenced-by-count":3,"title":["Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces"],"prefix":"10.1155","volume":"2017","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-0561","authenticated-orcid":true,"given":"Alexei Yu.","family":"Karlovich","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es, Departamento de Matem\u00e1tica, Faculdade de Ci\u00eancias e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829\u2013516 Caparica, Portugal"}]}],"member":"311","reference":[{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01564502"},{"key":"3","first-page":"89","volume":"213","year":"1964","journal-title":"Journal F\u00fcr Die Reine Und Angewandte Mathematik"},{"volume":"92","year":"2002","series-title":"Mathematical Surveys and Monographs","key":"23"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1070\/RM1982v037n01ABEH003192"},{"year":"2006","key":"2"},{"volume":"129","year":"1988","series-title":"Pure and Applied Mathematics","key":"1"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.5802\/aif.1313"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s00020-002-1190-z"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01192300"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(89)80007-1"},{"year":"1976","key":"13"},{"year":"1970","series-title":"Pure and Applied Mathematics, Vol. 38","key":"8"},{"year":"1991","key":"26"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3559-8"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1216\/jiea\/1181074970"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1973-0312139-8"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0548-3"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-18363-8"},{"first-page":"99","volume-title":"Hidegoro Nakano (1909\u20131974)\u2014on the centenary of his birth","year":"2011","key":"19"},{"volume":"248","year":"2016","series-title":"Operator Theory: Advances and Applications","key":"14"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1002\/mana.200510542"},{"key":"21","first-page":"439","volume":"8","year":"1960","journal-title":"Bulletin de l'Academie Polonaise des Sciences. Serie des Sciences Mathematiques, Astronomiques et Physiques"},{"key":"22","first-page":"11","volume":"26","year":"1988","journal-title":"Studia Mathematica"},{"volume":"234","journal-title":"Polska Akademia Nauk. Instytut Matematyczny. Dissertationes Mathematicae. Rozprawy Matematyczne","year":"1985","key":"17"},{"volume":"5","year":"1989","key":"18"},{"key":"12","series-title":"Operator Theory: Advances and Applications","first-page":"185","volume-title":"Singular integral operators on variable Lebesgue spaces with radial oscillating weights","volume":"195","year":"2010"}],"container-title":["Journal of Function Spaces"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2017\/9768210.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2017\/9768210.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jfs\/2017\/9768210.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,9,28]],"date-time":"2017-09-28T17:08:55Z","timestamp":1506618535000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jfs\/2017\/9768210\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"references-count":26,"alternative-id":["9768210","9768210"],"URL":"https:\/\/doi.org\/10.1155\/2017\/9768210","relation":{},"ISSN":["2314-8896","2314-8888"],"issn-type":[{"type":"print","value":"2314-8896"},{"type":"electronic","value":"2314-8888"}],"subject":[],"published":{"date-parts":[[2017]]}}}