{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:54Z","timestamp":1760240394470,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T00:00:00Z","timestamp":1559347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006469","name":"Macao Science and Technology Development Fund","doi-asserted-by":"publisher","award":["No.186\/2017\/A3"],"award-info":[{"award-number":["No.186\/2017\/A3"]}],"id":[{"id":"10.13039\/501100006469","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 \u2264 p &lt; \u221e ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( \u2202 D ) is reducing under the shift operator M e i \u03b8 on L 2 ( \u2202 D ) if and only if it is hyperinvariant under M e i \u03b8 . At the same time, we show that there are two large classes of invariant subspaces for M e i \u03b8 that are not hyperinvariant subspaces for M e i \u03b8 and are also not reducing subspaces for M e i \u03b8 . Moreover, we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.<\/jats:p>","DOI":"10.3390\/sym11060743","type":"journal-article","created":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T02:08:40Z","timestamp":1559527720000},"page":"743","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Invariant Subspaces for the Shift Operator"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9799-9671","authenticated-orcid":false,"given":"Junfeng","family":"Liu","sequence":"first","affiliation":[{"name":"Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau 999078, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,1]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Abramovich, Y.A., and Aliprantis, C.D. (2002). An Invitation to Operator Theory, American Mathematical Society.","key":"ref_1","DOI":"10.1090\/gsm\/050"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/j.jfa.2003.12.004","article-title":"Invariant subspaces for polynomially bounded operators","volume":"213","author":"Ambrozie","year":"2004","journal-title":"J. Funct. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11511-011-0058-y","article-title":"A hereditarily indecomposable \u2112\u221e -space that solves the scalar-plus-compact problem","volume":"206","author":"Argyros","year":"2011","journal-title":"Acta Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"345","DOI":"10.2307\/1969637","article-title":"Invariant subspaces of completely continuous operators","volume":"60","author":"Aronszajn","year":"1954","journal-title":"Ann. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s00020-013-2053-5","article-title":"Weighted Bergman spaces: Shift-invariant subspaces and input\/state\/output linear systems","volume":"76","author":"Ball","year":"2013","journal-title":"Integr. Equ. Oper. Theory"},{"doi-asserted-by":"crossref","unstructured":"Beauzamy, B. (1988). Introduction to Operator Theory and Invariant Subspaces, North-Holland.","key":"ref_6","DOI":"10.1016\/S0924-6509(08)70554-3"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"421","DOI":"10.2140\/pjm.1966.16.421","article-title":"Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos","volume":"16","author":"Bernstein","year":"1966","journal-title":"Pac. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/BF02395019","article-title":"On two problems concerning linear transformations in a Hilbert space","volume":"81","author":"Beurling","year":"1949","journal-title":"Acta Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1007\/BF01682842","article-title":"Some invariant subspaces for subnormal operators","volume":"1","author":"Brown","year":"1978","journal-title":"Integr. Equ. Oper. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"93","DOI":"10.2307\/1971289","article-title":"Hyponormal operators with thick spectra have invariant subspaces","volume":"125","author":"Brown","year":"1987","journal-title":"Ann. Math."},{"unstructured":"Duren, P.L. (2000). Theorey of Hp Spacce, Dover Publications Inc.","key":"ref_11"},{"doi-asserted-by":"crossref","unstructured":"Duren, P.L., and Schuster, A. (2004). Bergman Spaces, Math. Surveys Monographs, American Mathematical Society.","key":"ref_12","DOI":"10.1090\/surv\/100"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF02392260","article-title":"On the invariant subspace problem for Banach spaces","volume":"158","author":"Enflo","year":"1987","journal-title":"Acta Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/0022-1236(90)90034-I","article-title":"Invariant subspaces for operators with Bishop\u2019s property (\u03b2) and thick spectrum","volume":"94","author":"Eschmeier","year":"1990","journal-title":"J. Funt. Anal."},{"unstructured":"Eschmeier, J. (2003). Introduction to Banach Algebras, Operators, and Harmonic Analysis-Invariant Spaces, Cambridge University Press.","key":"ref_15"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"433","DOI":"10.2140\/pjm.1966.16.433","article-title":"Invariant subspaces of polynomially compact operators","volume":"16","author":"Halmos","year":"1966","journal-title":"Pac. J. Math."},{"unstructured":"Halmos, P.R. (1982). A Hilbert Space Problem Book, Springer-Verlag. [2nd ed.].","key":"ref_17"},{"unstructured":"Hoffman, K. (1988). Banach Spaces of Analytic Functions, Dover Publications Inc.","key":"ref_18"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.exmath.2006.11.003","article-title":"On the proof of Beurling\u2019s theorem on z-invariant subspaces","volume":"25","author":"Karaev","year":"2007","journal-title":"Expos. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s00020-008-1607-4","article-title":"On a Beurling-Arveson type theorem for some functional Hilbert spaces and related questions","volume":"62","author":"Karaev","year":"2008","journal-title":"Integr. Equ. Oper. Theory"},{"unstructured":"Koosis, P. (1998). Introduction to Hp spaces, Cambridge University Press.","key":"ref_21"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.21136\/CMJ.2017.0459-14","article-title":"On invariant subspaces for polynomially bounded operators","volume":"67","author":"Liu","year":"2017","journal-title":"Czechoslov. Math. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/BF01080698","article-title":"Invariant subspaces of the family of operators that commute with a completely continuous operator","volume":"7","author":"Lomonosov","year":"1973","journal-title":"Funct. Anal. Appl."},{"unstructured":"Mart\u00ednez-Avenda\u00f1o, R.A., and Rosenthal, P. (2007). An Introduction to Operators on the Hardy-Hilbert Space, Springer-Verlag.","key":"ref_24"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/0001-8708(77)90089-5","article-title":"Hilden\u2019s simple proof of Lomonosov\u2019s invariant subspace theorem","volume":"25","author":"Michaels","year":"1977","journal-title":"Adv. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jfa.2013.04.002","article-title":"Every operator has almost-invariant subspaces","volume":"265","author":"Popov","year":"2013","journal-title":"J. Funct. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"683","DOI":"10.2307\/2373347","article-title":"On invariant subspaces and reflexive algebras","volume":"91","author":"Radjavi","year":"1969","journal-title":"Am. J. Math."},{"unstructured":"Radjavi, H., and Rosenthal, P. (2003). Invariant Subspaces, Dover Publications Inc.. [2nd ed.].","key":"ref_28"},{"key":"ref_29","first-page":"437","article-title":"Invariant sublattices","volume":"52","author":"Radjavi","year":"2008","journal-title":"Ill. J. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1112\/blms\/16.4.337","article-title":"A solution to the invariant subspace problem","volume":"16","author":"Read","year":"1984","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1112\/blms\/17.4.305","article-title":"A solution to the invariant subspace problem on the space l1","volume":"17","author":"Read","year":"1985","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1112\/jlms\/s2-34.2.335","article-title":"A short proof concerning the invariant subspace problem","volume":"34","author":"Read","year":"1986","journal-title":"J. Lond. Math. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1112\/S0024610797005486","article-title":"Quasinilpotent operators and the invariant subspace problem","volume":"56","author":"Read","year":"1997","journal-title":"J. Lond. Math. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1007\/s10688-017-0173-4","article-title":"On operators with orbits dense relative to nontrivial subspaces","volume":"51","author":"Rezaei","year":"2017","journal-title":"Funct. Anal. Its Appl."},{"unstructured":"Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill Companies, Inc.. [3rd ed.].","key":"ref_35"},{"key":"ref_36","first-page":"147","article-title":"Wold-type decompositions and wandering subspaces for operators close to isometries","volume":"531","author":"Shimorin","year":"2001","journal-title":"J. Reine Angew. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1090\/S0002-9939-02-06721-7","article-title":"On Beurling-type theorems in weighted l2 and Bergman Spaces","volume":"131","author":"Shimorin","year":"2002","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/743\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:32Z","timestamp":1760187332000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/6\/743"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,1]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["sym11060743"],"URL":"https:\/\/doi.org\/10.3390\/sym11060743","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,6,1]]}}}