{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:19:56Z","timestamp":1772507996029,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>If T is a bounded linear operator on a Hilbert space H and V is a given linear isometry on a Hilbert space K, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry \u03c0:H\u2192K such that \u03c0T=V*\u03c0 (i.e., (\u03c0,V*) extends T). We parametrize the set of all solutions \u03c0 of this equation. We show, for example, that for a given unitary operator U on a Hilbert space E and for the multiplication operator by the independent variable Mz on the Hardy space HD2(D), there exists an isometric operator \u03c0:H\u2192E\u2295HD2(D) such that (\u03c0,(U\u2295Mz)*) extends T if and only if T is a contraction, the defect index \u03b4T\u2264dimD and, for some Y:AT\u2192E, (Y,U*) extends the isometric operator AT1\/2h\u21a6AT1\/2Th on the space AT=ATH\u00af, where AT is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator \u03c0:H\u2192K such that (\u03c0,V) extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces K0 possess identical multiplicity functions and (b) dim(kerT*)=dim(K1\u2296VK1) for a certain subspace K1 of K that contains K0 and is invariant under V. The precise form of \u03c0, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes.<\/jats:p>","DOI":"10.3390\/axioms12020202","type":"journal-article","created":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T01:38:18Z","timestamp":1676425098000},"page":"202","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Structure of Coisometric Extensions"],"prefix":"10.3390","volume":"12","author":[{"given":"Dan","family":"Popovici","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, West University of Timi\u0219oara, Vasile P\u00e2rvan Blvd No. 4, RO-300223 Timi\u0219oara, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"ref_1","unstructured":"Reed, M., and Simon, B. (1980). Methods of Modern Mathematical Physics, Academic Press. Functional Analysis."},{"key":"ref_2","unstructured":"Riesz, F., and Sz.-Nagy, B. (1955). Functional Analysis, Ungar."},{"key":"ref_3","unstructured":"Rudin, W. (1991). Functional Analysis, McGraw-Hill. [8th ed.]. International Series in Pure and Applied Mathematics."},{"key":"ref_4","first-page":"87","article-title":"Sur les contractions de l\u2019espace de Hilbert","volume":"15","year":"1953","journal-title":"Acta Sci. Math. Szeged"},{"key":"ref_5","first-page":"551","article-title":"Dilation Theory: A Guided Tour. Operator Theory, Functional Analysis and Applications","volume":"282","author":"Shalit","year":"2021","journal-title":"Oper. Th. Adv. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sz.-Nagy, B., Foia\u0219, C., Bercovici, H., and K\u00e9rchy, L. (2010). Harmonic Analysis of Operators on Hilbert Space, Springer. [2nd ed.]. Revised and Enlarged edition.","DOI":"10.1007\/978-1-4419-6094-8"},{"key":"ref_7","unstructured":"Foia\u0219, C., and Frazho, A.E. (1990). Operator Theory: Advances and Applications, Birkh\u00e4user-Verlag."},{"key":"ref_8","unstructured":"Foia\u0219, C., Frazho, A.E., Gohberg, I., and Kaashoek, M.A. (1998). Operator Theory: Advances and Applications, Birkh\u00e4user-Verlag."},{"key":"ref_9","unstructured":"Rosenblum, M., and Rovnyak, J. (1985). Oxford Mathematical Monographs, Oxford University Press."},{"key":"ref_10","unstructured":"Kakihara, Y. (1997). Series on Multivariate Analysis 2, World Scientific Publishing Co., Inc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1090\/conm\/414\/07823","article-title":"Extensions and dilations for C*-dynamical systems. Operator Theory, Operator Algebras and Applications","volume":"414","author":"Muhly","year":"2006","journal-title":"Contemp. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1006\/jfan.1998.3294","article-title":"Tensor algebras over C*-correspondences: Representations, dilations, and C*-envelopes","volume":"158","author":"Muhly","year":"1998","journal-title":"J. Funct. Anal."},{"key":"ref_13","unstructured":"Wolf, T. (2013). Coisometric Extensions. [Ph.D. Thesis, University of Iowa]."},{"key":"ref_14","first-page":"12","article-title":"Sur les contractions de l\u2019espace de Hilbert. VII. Triangulations canoniques. Fonctions minimum","volume":"25","year":"1964","journal-title":"Acta Sci. Math. Szeged"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"443","DOI":"10.7900\/jot.2019nov02.2273","article-title":"Factors of hypercontractions","volume":"85","author":"Bhattacharjee","year":"2021","journal-title":"J. Oper. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.aim.2017.10.010","article-title":"Factorizations of contractions","volume":"322","author":"Das","year":"2017","journal-title":"Adv. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1515\/crll.1961.208.102","article-title":"Shifts on Hilbert spaces","volume":"208","author":"Halmos","year":"1961","journal-title":"J. Reine Angew. Math."},{"key":"ref_18","unstructured":"Halmos, P.R. (2017). Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Dover Publications. [2nd ed.]."},{"key":"ref_19","first-page":"349","article-title":"A remark on the universal model for contractions of G. C. Rota","volume":"13","year":"1963","journal-title":"Com. Acad. R. P. Rom\u00eene"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1090\/S0002-9939-1955-0068740-7","article-title":"On unitary dilations of contractions","volume":"6","year":"1955","journal-title":"Proc. Amer. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1215\/S0012-7094-61-02856-3","article-title":"The unitary dilation of a contraction operator","volume":"28","author":"Halperin","year":"1961","journal-title":"Duke Math. J."},{"key":"ref_22","first-page":"343","article-title":"On Sch\u00e4ffer\u2019s construction of unitary dilations","volume":"3","year":"1960","journal-title":"Ann. Univ. Budapest Sect. Math."},{"key":"ref_23","first-page":"106","article-title":"Sur les contractions de l\u2019espace de Hilbert. V. Translation bilat\u00e9rales","volume":"23","year":"1962","journal-title":"Acta Sci. Math. Szeged"},{"key":"ref_24","first-page":"17","article-title":"Spectral behaviour of C10-contractions","volume":"Volume 12","author":"Bercovici","year":"2010","journal-title":"Operator Theory Live"},{"key":"ref_25","first-page":"49","article-title":"Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems","volume":"53","author":"Cassier","year":"2005","journal-title":"J. Oper. Theory"},{"key":"ref_26","first-page":"593","article-title":"Contractions with C\u00b70 direct summands","volume":"11","author":"Kubrusly","year":"2001","journal-title":"Adv. Math. Sci. Appl."},{"key":"ref_27","first-page":"1243","article-title":"On unitary parts of contractions","volume":"25","author":"Duggal","year":"1994","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"603","DOI":"10.14232\/actasm-013-255-6","article-title":"Contractions T for which A is a projection","volume":"80","author":"Kubrusly","year":"2014","journal-title":"Acta Sci. Math. Szeged"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kubrusly, C.S. (2003). Hilbert Space Operators: A Problem Solving Approach, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-2064-0"},{"key":"ref_30","first-page":"123","article-title":"Strong stability for cohyponormal operators","volume":"31","author":"Kubrusly","year":"1994","journal-title":"J. Operator Theory"},{"key":"ref_31","first-page":"523","article-title":"A decomposition for a class of contractions","volume":"6","author":"Kubrusly","year":"1996","journal-title":"Adv. Math. Sci. Appl."},{"key":"ref_32","unstructured":"Wold, H. (1938). A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Xuan, A., Yin, M., Li, Y., Chen, X., and Ma, Z. (2022, January 1\u20133). A comprehensive evaluation of statistical, machine learning and deep learning models for time series prediction. Proceedings of the 7th International Conference on Data Science and Machine Learning Applications (CDMA), Virtual.","DOI":"10.1109\/CDMA54072.2022.00014"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1061\/(ASCE)0733-947X(2003)129:6(664)","article-title":"Modeling and forecasting vehicular traffic flow as a seasonal ARIMA process: Theoretical basis and empirical results","volume":"129","author":"Williams","year":"2003","journal-title":"J. Transp. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"108680","DOI":"10.1016\/j.ymssp.2021.108680","article-title":"Cyclostationary modeling for the aerodynamically generated sound of helicopter rotors","volume":"168","author":"Yu","year":"2022","journal-title":"Data Mech. Syst. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1109\/RBME.2021.3115703","article-title":"Texture Analysis and Its Applications in Biomedical Imaging: A Survey","volume":"15","author":"Ghalati","year":"2022","journal-title":"IEEE Rev. Biomed. Eng."},{"key":"ref_37","first-page":"1","article-title":"Stationary sequences in Hilbert spaces","volume":"2","author":"Kolmogorov","year":"1941","journal-title":"Bull. Math. Univ. Moskow"},{"key":"ref_38","first-page":"49","article-title":"Allgemeine eigenwerttheorie hermitischer funktional operatoren","volume":"102","year":"1929","journal-title":"Math. Ann."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1090\/surv\/013\/03","article-title":"A version of multiplicity theory","volume":"Volume 13","author":"Brown","year":"1974","journal-title":"Topics in Operator Theory"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Kubrusly, C.S. (1997). An Introduction to Models and Decompositions in Operator Theory, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1998-9"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1090\/tran\/7676","article-title":"Isometric dilations and von Neumann inequality for a class of tuples in the polydisc","volume":"372","author":"Barik","year":"2019","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_42","unstructured":"Sarkar, S. (2023). Pairs of Commuting Pure Contractions and Isometric dilation. J. Operator Theory, accepted."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1512\/iumj.1993.42.42035","article-title":"Standard Operators Models in the Polydisc I","volume":"42","author":"Curto","year":"1993","journal-title":"Indiana Univ. Math. J."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"297","DOI":"10.4064\/sm170-3-6","article-title":"On C0\u00b7 multi-contractions having a regular dilation","volume":"170","author":"Popovici","year":"2005","journal-title":"Studia Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/202\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:35:34Z","timestamp":1760121334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/202"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,14]]},"references-count":44,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020202"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020202","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,14]]}}}