{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:48:08Z","timestamp":1747198088803,"version":"3.40.5"},"reference-count":2,"publisher":"Walter de Gruyter GmbH","issue":"5","funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/04459\/2019","UIDB\/00212\/2020","UIDB\/04674\/2020"],"award-info":[{"award-number":["UID\/MAT\/04459\/2019","UIDB\/00212\/2020","UIDB\/04674\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,9,1]]},"abstract":"Abstract<\/jats:title>\n In this note, we characterize the essential numerical range of a block diagonal operator \n \n \n \n T<\/m:mi>\n =<\/m:mo>\n \n \n \u2295<\/m:mo>\n i<\/m:mi>\n <\/m:msub>\n \n T<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {T=\\bigoplus_{i}T_{i}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in terms of the numerical ranges \n \n \n \n \n {<\/m:mo>\n \n W<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n T<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n i<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {\\{W(T_{i})\\}_{i}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of its components. Specifically, the essential numerical range of T<\/jats:italic> is the convex hull of the limit superior of \n \n \n \n \n {<\/m:mo>\n \n W<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n T<\/m:mi>\n i<\/m:mi>\n <\/m:msub>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n i<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {\\{W(T_{i})\\}_{i}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. This characterization can be simplified further. In fact, we prove the existence of a decomposition of T<\/jats:italic> for which the convex hull is not required.<\/jats:p>","DOI":"10.1515\/forum-2023-0166","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T12:46:57Z","timestamp":1701262017000},"page":"1147-1157","source":"Crossref","is-referenced-by-count":1,"title":["A note on the essential numerical range of block diagonal operators"],"prefix":"10.1515","volume":"36","author":[{"given":"Lu\u00eds","family":"Carvalho","sequence":"first","affiliation":[{"name":"ISCTE , Lisbon University Institute , Av. das For\u00e7as Armadas, 1649-026 , Lisbon ; and Center for Mathematical Analysis, Geometry, and Dynamical Systems Mathematics Department, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa , Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9876-7626","authenticated-orcid":false,"given":"Cristina","family":"Diogo","sequence":"additional","affiliation":[{"name":"ISCTE , Lisbon University Institute , Av. das For\u00e7as Armadas, 1649-026 , Lisbon ; and Center for Mathematical Analysis, Geometry, and Dynamical Systems Mathematics Department, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa , Portugal"}]},{"given":"S\u00e9rgio","family":"Mendes","sequence":"additional","affiliation":[{"name":"ISCTE , Lisbon University Institute , Av. das For\u00e7as Armadas, 1649-026 , Lisbon ; and Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es, Universidade da Beira Interior, Rua Marqu\u00eas d\u2019\u00c1vila e Bolama, 6201-001, Covilh\u00e3 , Portugal"}]},{"given":"Helena","family":"Soares","sequence":"additional","affiliation":[{"name":"ISCTE , Lisbon University Institute , Av. das For\u00e7as Armadas, 1649-026 , Lisbon ; and Centro de Investiga\u00e7\u00e3o em Matem\u00e1tica e Aplica\u00e7\u00f5es, Universidade de \u00c9vora, Col\u00e9gio Lu\u00eds Ant\u00f3nio Verney, Rua Rom\u00e3o Ramalho, 59, 7000-671 \u00c9vora , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2023,11,30]]},"reference":[{"key":"2024082213230505371_j_forum-2023-0166_ref_001","doi-asserted-by":"crossref","unstructured":"G. D. Allen and J. D. Ward,\nA simultaneous lifting theorem for block diagonal operators,\nTrans. Amer. Math. Soc. 250 (1979), 385\u2013397.","DOI":"10.1090\/S0002-9947-1979-0530063-8"},{"key":"2024082213230505371_j_forum-2023-0166_ref_002","unstructured":"P. A. Fillmore, J. G. Stampfli and J. P. Williams,\nOn the essential numerical range, the essential spectrum, and a problem of Halmos,\nActa Sci. Math. (Szeged) 33 (1972), 179\u2013192."}],"container-title":["Forum Mathematicum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2023-0166\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2023-0166\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T21:22:01Z","timestamp":1724361721000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/forum-2023-0166\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,30]]},"references-count":2,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,12,15]]},"published-print":{"date-parts":[[2024,9,1]]}},"alternative-id":["10.1515\/forum-2023-0166"],"URL":"https:\/\/doi.org\/10.1515\/forum-2023-0166","relation":{},"ISSN":["0933-7741","1435-5337"],"issn-type":[{"type":"print","value":"0933-7741"},{"type":"electronic","value":"1435-5337"}],"subject":[],"published":{"date-parts":[[2023,11,30]]}}}