{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T18:18:48Z","timestamp":1775413128246,"version":"3.50.1"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"7","license":[{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"content-version":"vor","delay-in-days":365,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc. Ser. B"],"abstract":"

\n Let\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a rearrangement-invariant Banach function space on the unit circle\n \n \n \n \n T<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {T}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and let\n \n \n \n \n H<\/mml:mi>\n [<\/mml:mo>\n X<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n H[X]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the abstract Hardy space built upon\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove that if the Cauchy singular integral operator\n \n \n \n \n (<\/mml:mo>\n H<\/mml:mi>\n f<\/mml:mi>\n )<\/mml:mo>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n 1<\/mml:mn>\n \n \n \u03c0\n \n <\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n \u222b\n \n <\/mml:mo>\n \n \n T<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n f<\/mml:mi>\n (<\/mml:mo>\n \n \u03c4\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n \u03c4\n \n <\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n d<\/mml:mi>\n \n \u03c4\n \n <\/mml:mi>\n <\/mml:mrow>\n (Hf)(t)=\\frac {1}{\\pi i}\\int _{\\mathbb {T}}\\frac {f(\\tau )}{\\tau -t}\\,d\\tau<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is bounded on the space\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then the norm, the essential norm, and the Hausdorff measure of non-compactness of the operator\n \n \n \n \n a<\/mml:mi>\n I<\/mml:mi>\n +<\/mml:mo>\n b<\/mml:mi>\n H<\/mml:mi>\n <\/mml:mrow>\n aI+bH<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n a<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n a,b\\in \\mathbb {C}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , acting on the space\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , coincide. We also show that similar equalities hold for the backward shift operator\n \n \n \n \n (<\/mml:mo>\n S<\/mml:mi>\n f<\/mml:mi>\n )<\/mml:mo>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n (<\/mml:mo>\n f<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \n f<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n (<\/mml:mo>\n 0<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n t<\/mml:mi>\n <\/mml:mrow>\n (Sf)(t)=(f(t)-\\widehat {f}(0))\/t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on the abstract Hardy space\n \n \n \n \n H<\/mml:mi>\n [<\/mml:mo>\n X<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n H[X]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Our results extend those by Krupnik and Polonski\u012d [Funkcional. Anal. i Priloz\u0306en. 9 (1975), pp. 73-74] for the operator\n \n \n \n \n a<\/mml:mi>\n I<\/mml:mi>\n +<\/mml:mo>\n b<\/mml:mi>\n H<\/mml:mi>\n <\/mml:mrow>\n aI+bH<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and by the second author [J. Funct. Anal. 280 (2021), p. 11] for the operator\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/bproc\/118","type":"journal-article","created":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T16:08:56Z","timestamp":1647965336000},"page":"60-70","source":"Crossref","is-referenced-by-count":6,"title":["On the essential norms of singular integral operators with constant coefficients and of the backward shift"],"prefix":"10.1090","volume":"9","author":[{"given":"Oleksiy","family":"Karlovych","sequence":"first","affiliation":[]},{"given":"Eugene","family":"Shargorodsky","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,3,22]]},"reference":[{"key":"1","series-title":"Operator Theory: Advances and Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5727-7","volume-title":"Measures of noncompactness and condensing operators","volume":"55","author":"Akhmerov, R. 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