{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,23]],"date-time":"2025-09-23T13:17:02Z","timestamp":1758633422823,"version":"3.37.3"},"reference-count":13,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,11,25]],"date-time":"2018-11-25T00:00:00Z","timestamp":1543104000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Daiichi University of Pharmacy"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2018,11,25]]},"abstract":"<jats:p>A sequence <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> in a Hilbert space <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi mathvariant=\"script\">H<\/mml:mi><\/mml:mrow><\/mml:math> with inner product <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mo>&lt;<\/mml:mo><mml:mo>\u00b7<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u00b7<\/mml:mo><mml:mo>&gt;<\/mml:mo><\/mml:math> is called a generalized Riesz system if there exist an ONB <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi mathvariant=\"bold-italic\">e<\/mml:mi><mml:mo>=<\/mml:mo><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi mathvariant=\"script\">H<\/mml:mi><\/mml:mrow><\/mml:math> and a densely defined closed operator <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:math> in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi mathvariant=\"script\">H<\/mml:mi><\/mml:mrow><\/mml:math> with densely defined inverse such that <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><mml:mo>\u2282<\/mml:mo><mml:mi>D<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\u2229<\/mml:mo><mml:mi>D<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mo>\u204e<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>T<\/mml:mi><mml:msub><mml:mrow><mml:mi>e<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>n<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">0,1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mo>\u22ef<\/mml:mo><\/mml:math>, and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"bold-italic\">e<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>T<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> is called a constructing pair for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><\/mml:math> is called a constructing operator for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math>. The main purpose of this paper is to investigate under what conditions the ordered set <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:msub><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> of all constructing operators for a generalized Riesz system <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mo stretchy=\"false\">{<\/mml:mo><mml:msub><mml:mrow><mml:mi>\u03c6<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">}<\/mml:mo><\/mml:math> has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications.<\/jats:p>","DOI":"10.1155\/2018\/3268251","type":"journal-article","created":{"date-parts":[[2018,11,25]],"date-time":"2018-11-25T18:30:55Z","timestamp":1543170655000},"page":"1-8","source":"Crossref","is-referenced-by-count":4,"title":["Ordered Structures of Constructing Operators for Generalized Riesz Systems"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5253-5144","authenticated-orcid":true,"given":"Hiroshi","family":"Inoue","sequence":"first","affiliation":[{"name":"Center for Advancing Pharmaceutical Education, Daiichi University of Pharmacy, 22-1 Tamagawa-cho, Minami-ku, Fukuoka 815-8511, Japan"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5020427"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1063\/1.4960476"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1063\/1.4966927"},{"year":"2002","key":"4"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/aa60ff"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/1564"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1142\/S0219887810004816"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/49\/40\/405202"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2017.01.079"},{"year":"1990","key":"10"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s13538-011-0010-9"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2010.07.044"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1063\/1.1640796"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/3268251.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/3268251.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2018\/3268251.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,11,25]],"date-time":"2018-11-25T18:30:58Z","timestamp":1543170658000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2018\/3268251\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,25]]},"references-count":13,"alternative-id":["3268251","3268251"],"URL":"https:\/\/doi.org\/10.1155\/2018\/3268251","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2018,11,25]]}}}