{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T11:12:02Z","timestamp":1760181122720,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T00:00:00Z","timestamp":1605830400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.<\/jats:p>","DOI":"10.3390\/sym12111918","type":"journal-article","created":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T11:50:34Z","timestamp":1606132234000},"page":"1918","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9991-5547","authenticated-orcid":false,"given":"Oleh","family":"Lopushansky","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Rzesz\u00f3w, 35-310 Rzesz\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Renata","family":"T\u0142uczek-Pi\u0229ciak","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Rzesz\u00f3w, 35-310 Rzesz\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bergh, J., and L\u00f6fstr\u00f6m, J. (1976). Interpolation Spaces, Springer.","DOI":"10.1007\/978-3-642-66451-9"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/BF02417949","article-title":"Interpolation of normed Abelian groups","volume":"92","author":"Peetre","year":"1972","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s11253-005-0225-4","article-title":"Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method","volume":"57","author":"Gorbachuk","year":"2005","journal-title":"Ukrainian Math. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3659","DOI":"10.1007\/s11785-019-00923-0","article-title":"On Spectral Approximations of Unbounded Operators","volume":"13","author":"Dmytryshyn","year":"2019","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_5","first-page":"1","article-title":"Bernstein-Jackson-type inequalities and Besov spaces associated with unbounded operators","volume":"105","author":"Dmytryshyn","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_6","unstructured":"Triebel, H. (1978). Interpolation Theory. Function Spaces. Differential Operators, North-Holland Publ."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Birman, M.S., and Solomjak, M.Z. (1987). Spectral Theory of Self-Adjoint Operators in Hilbert Space, Springer.","DOI":"10.1007\/978-94-009-4586-9"},{"key":"ref_8","unstructured":"McLean, W. (2000). Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"383","DOI":"10.2748\/tmj\/1178229401","article-title":"A general interpolation theorem of Marcinkiewicz type","volume":"33","author":"Komatsu","year":"1981","journal-title":"T\u00f4hoku Math. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1016\/j.jmaa.2015.07.004","article-title":"Monotonicity properties and dominated best approximation problems in Orlicz spaces equipped with the p-Amemiya norm","volume":"432","author":"Cui","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Yang, Y. (2019). Viscosity approximation methods for zeros of accretive operators. J. Nonlinear Funct. Anal.","DOI":"10.23952\/jnfa.2019.49"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1918\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:35:15Z","timestamp":1760178915000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/11\/1918"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,20]]},"references-count":11,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["sym12111918"],"URL":"https:\/\/doi.org\/10.3390\/sym12111918","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,11,20]]}}}