{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T17:37:21Z","timestamp":1764092241514,"version":"3.45.0"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T00:00:00Z","timestamp":1763942400000},"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>The concept of fractional powers of operators was initially developed within the framework of functional analysis and has since played a significant role in the study of evolution equations, abstract differential equations, and complex dynamical systems. Meanwhile, the operator matrix approach has emerged as a powerful and widely used tool for analyzing the structural properties of operators. Motivated by these developments, this paper focuses on the explicit computation of fractional powers of anti-triangular operator matrices. Specifically, we first derive an explicit formula for the fractional powers of off-diagonal operator matrices by employing the formal determinant of block operator matrices. Then, based on the Schur factorization, we obtain a representation for the fractional powers of the general anti-triangular case. As applications, the obtained results are further applied to certain differential equations.<\/jats:p>","DOI":"10.3390\/axioms14120859","type":"journal-article","created":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T17:31:34Z","timestamp":1764091894000},"page":"859","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fractional Powers of Anti-Triangular Operator Matrices and Their Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-4215-8353","authenticated-orcid":false,"given":"Lei","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]},{"given":"Yongqing","family":"Fu","sequence":"additional","affiliation":[{"name":"School of Methematics and Computer Engineering, Ordos Institute of Technology, Ordos 017000, China"}]},{"given":"Junjie","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Amann, H.H. (1995). Linear and Quasilinear Parabolic Problems, Birkh\u00e4user.","DOI":"10.1007\/978-3-0348-9221-6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"419","DOI":"10.2140\/pjm.1960.10.419","article-title":"Fractional powers of closed operators and the semigroups generated by them","volume":"10","author":"Balakrishnan","year":"1960","journal-title":"Pac. J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3288","DOI":"10.1002\/mana.202300318","article-title":"On coupled semilinear evolution systems: Techniques on fractional powers of 4\u00d74 matrices and applications","volume":"297","author":"Belluzi","year":"2024","journal-title":"Math. Nachr."},{"key":"ref_4","unstructured":"Martinez Carracedo, C., and Sanz Alix, M. (2001). The Theory of Fractional Powers of Operators, Elsevier."},{"key":"ref_5","unstructured":"Hille, E., and Phillips, R.S. (1974). Functional Analysis and Semi-Groups, American Mathematical Society."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"285","DOI":"10.2140\/pjm.1966.19.285","article-title":"Fractional powers of operators","volume":"19","author":"Komatsu","year":"1966","journal-title":"Pac. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"89","DOI":"10.2140\/pjm.1967.21.89","article-title":"Fractional powers of operators. II. Interpolation spaces","volume":"21","author":"Komatsu","year":"1967","journal-title":"Pac. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"246","DOI":"10.2969\/jmsj\/01330246","article-title":"Fractional powers of dissipative operators","volume":"13","author":"Kato","year":"1961","journal-title":"J. Math. Soc. Japan"},{"key":"ref_9","first-page":"94","article-title":"Note on fractional powers of linear operators","volume":"36","author":"Kato","year":"1960","journal-title":"Proc. Japan Acad."},{"key":"ref_10","first-page":"499","article-title":"Fractional powers of operators acting in Banach spaces","volume":"129","author":"Sobolevskii","year":"1959","journal-title":"Doklady Akad. Nauk SSSR"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"343","DOI":"10.2140\/pjm.1952.2.343","article-title":"On the generation of semi-groups of linear operators","volume":"2","author":"Phillips","year":"1952","journal-title":"Pac. J. Math."},{"key":"ref_12","first-page":"273","article-title":"On some properties of fractional powers of linear operators","volume":"37","author":"Watanabe","year":"1961","journal-title":"Proc. Japan Acad."},{"key":"ref_13","first-page":"86","article-title":"Fractional powers of infinitesimal generators and the analyticity of the semi-groups generated by them","volume":"36","author":"Yosida","year":"1960","journal-title":"Proc. Jpn. Acad."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"319","DOI":"10.3934\/nhm.2017014","article-title":"Exact and positive controllability of boundary control systems","volume":"12","author":"Engel","year":"2017","journal-title":"Netw. Heterog. Media"},{"key":"ref_15","first-page":"61","article-title":"Operator matrices and systems of evolution equations","volume":"966","author":"Engel","year":"1996","journal-title":"RIMS Kokyuroku"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF02925301","article-title":"Operator matrices and reaction-difusion systems","volume":"59","author":"Nagel","year":"1989","journal-title":"Rend. Sem. Mat. Fis. Milano"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1002\/mana.19941670102","article-title":"The essential spectrum of some matrix operators","volume":"167","author":"Atkinson","year":"1994","journal-title":"Math. Nachr."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"125644","DOI":"10.1016\/j.jmaa.2021.125644","article-title":"On a cascade system of Schrodinger equations. Fractional powers approach","volume":"506","author":"Belluzi","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","unstructured":"Baloudi, H., Dbeibia, M.A., and Jeribi, A. (2025). Fractional powers of block operator matrices: Applications to PDEs. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/j.jmaa.2017.01.024","article-title":"Parabolic approximation of damped wave equations via fractional powers: Fast growing nonlinearities and continuity of the dynamics","volume":"450","author":"Bezerra","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4221","DOI":"10.3934\/dcdsb.2020095","article-title":"Fractional approximations of abstract semilinear parabolic problems","volume":"25","author":"Bezerra","year":"2020","journal-title":"Discrete Contin. Dyn. Syst. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5661","DOI":"10.1016\/j.jde.2020.04.020","article-title":"Fractional powers approach of operators for abstract evolution equations of third order in time","volume":"269","author":"Bezerra","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_23","first-page":"319","article-title":"Some problems for linear elastic systems with damping","volume":"3","author":"Huang","year":"1990","journal-title":"Acta Math. Sin. Ser. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0252-9602(18)30548-4","article-title":"On the holomorphic property of the semigroup associated with linear elastic systems with structural damping","volume":"5","author":"Huang","year":"1985","journal-title":"Acta Math. Sci."},{"key":"ref_25","first-page":"196","article-title":"The representation of the Drazin inverse of anti-triangular operator matrices based on resolvent expansions","volume":"242","author":"Huang","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.cam.2023.115187","article-title":"Exact expressions for the Drazin inverse of anti-triangular matrices","volume":"428","author":"Zhang","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"ref_27","unstructured":"Engel, K.-J., and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer. Graduate Texts in Mathematics."},{"key":"ref_28","unstructured":"Kato, T. (2013). Perturbation Theory for Linear Operators, Springer Science and Business Media."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1982). A Hilbert Space Problem Book, Springer. [2nd ed.]. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4684-9330-6"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Respondek, J.S. (2025). Fast Matrix Multiplication with Applications, Springer.","DOI":"10.1007\/978-3-031-76930-6"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/859\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T17:32:20Z","timestamp":1764091940000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/859"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,24]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["axioms14120859"],"URL":"https:\/\/doi.org\/10.3390\/axioms14120859","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,24]]}}}