Publication:
Multiplication by a finite Blaschke product on weighted Bergman spaces: Commutant and reducing subspaces

dc.contributor.authorGallardo-Gutiérrez, Eva A.
dc.contributor.authorPartington, Johathan R.
dc.date.accessioned2023-06-22T10:48:58Z
dc.date.available2023-06-22T10:48:58Z
dc.date.issued2022
dc.description.abstractWe provide a characterization of the commutant of analytic Toeplitz operators TB induced by finite Blachke products B acting on weighted Bergman spaces which, as a particular instance, yields the case B(z) = z n on the Bergman space solved recently by by Abkar, Cao and Zhu [2]. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces Hp for 1 < p < ∞. Finally, we apply this approach to study reducing subspaces of TB in the classical Bergman space. As a particular instance, we provide a direct proof of a theorem of Hu, Sun, Xu and Yu [18] which states that every analytic Toeplitz operator TB induced by a finite Blachke product on the Bergman space is reducible and the restriction of TB on a reducing subspace is unitarily equivalent to the Bergman shift.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedFALSE
dc.description.sponsorshipPlan Nacional I+D
dc.description.sponsorshipPrograma para Centros de Excelencia en Investigación y Desarrollo Severo Ochoa
dc.description.sponsorshipAyuda extraordinaria a Centros de Excelencia Severo Ochoa
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/73300
dc.identifier.citation[1] M. Abrahamse and J. Ball, Analytic Toeplitz operators with automorphic symbol II, Proc. Amer. Math. Soc. 59, 323–328 (1976). [2] A. Abkar, G. Cao and K. Zhu, The commutant of some shift operators, Complex Analysis and Operator Theory, 14 (2020), no. 6, art. 58. [3] I. Chalendar, E.A. Gallardo-Guti´errez, and J.R. Partington, Weighted composition operators on the Dirichlet space: Boundedness and spectral properties, Math. Ann. 363 (2015) 1265–1279. [4] C. C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Am. Math. Soc. 239 (1978), 1–31 [5] C. C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Univ. Math. J. 29, 1–12 (1980). [6] C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators, J. Funct. Anal. 36, 169–184 (1980). [7] C. C. Cowen and R. G. Wahl, Commutants of finite Blaschke product multiplication operators, Invariant subspaces of the shift operator, 99–114, Contemp. Math., 638, Amer. Math. Soc., Providence, RI, 2015. [8] J. Deddens and T. Wong, The commutant of analytic Toeplitz operators Trans. Am. Math. Soc. 184, 261–273 (1973). [9] R. G. Douglas, S. Sun and D. Zheng, Multiplication operators on the Bergman space via analytic continuation, Adv. Math. 226 (2012), 541–583. [10] R. G. Douglas, M. Putinar and K. Wang, Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal. 263 (2012), 1744–1765. [11] P. Duren, Theory of H p spaces, Academic Press, New York (1970). [12] P. Duren, D. Khavinson, H.S. Shapiro and C. Sundberg, Contractive zero-divisors in Bergman spaces. Pacific J. Math. 157 (1993), no. 1, 37–56. [13] E.A. Gallardo-Guti´errez, J.R. Partington and D. Seco, On the Wandering Property in Dirichlet spaces, Integral Equations Operator Theory 92 (2020), no. 2, Paper No. 16. [14] K. Guo and H. Huang, On multiplication operators of the Bergman space: similarity, unitary equivalence and reducing subspaces, J. Operator Theory 65 (2011), 355–378. [15] K. Guo and H. Huang, Multiplication operators defined by covering maps on the Bergman space: the connection between operator theory and von Neumann algebras, J. Funct. Anal. 260 (2011), 1219–1255. [16] K. Guo and H. Huang, Geometric constructions of thin Blaschke products and reducing subspace problem, Proc. Lond. Math. Soc. 109 (2014), 1050–1091. [17] K. Guo and H. Huang, Multiplication operators on the Bergman space. Lecture Notes in Mathematics, 2145. Springer, Heidelberg, 2015. [18] J. Hu, S. Sun, X. Xu and D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space, Integral Equations Operator Theory 49 (2004), 387–395. [19] A.L. Shields, Weighted shift operators and analytic function theory. Topics in operator theory, 49–128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974. [20] A. Shields and L. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20, 777–788 (1970/1971). [21] M. Stessin and K. Zhu, Reducing subspace of weighted shift operators, Proc. Amer. Math. Soc., 130(2002), 2631–2639 [22] S.L. Sun and Y. Wang, Reducing subspaces of certain analytic Toeplitz operators on the Bergman space, Northeast. Math. J. 14, 147–158 (1998) [23] J. Thomson, The commutant of a class of analytic Toeplitz operators, American J. Math. 99, 522–529 (1977). [24] J. Thomson, The commutant of a class of analytic Toeplitz operators II, Indiana Univ. Math. J. 25, 793–800 (1976) [25] J. Thomson, The commutant of certain analytic Toeplitz operators, Proc. Amer. Math. Soc. 54, 165–169 (1976). [26] R. Zhao a d K. Zhu, Theory of Bergman spaces in the unit ball of Cn, M´em. Soc. Math. Fr. (N.S.) No. 115 (2008), vi+103 pp. [27] K. Zhu, Reducing subspaces for a class of multiplication operators, J. Lond. Math. Soc. 62, 553–568 (2000). [28] K. Zhu, Operator theory in function spaces. Second edition. Mathematical Surveys and Monographs, 138. American Mathematical Society, Providence, RI, 2007. [29] N. Zorboska, Isometric weighted composition operators on weighted Bergman spaces, J. Math. Anal. Appl. 461 (2018), no. 1, 657–675.
dc.identifier.issn0022-247X
dc.identifier.urihttps://hdl.handle.net/20.500.14352/71718
dc.issue.number1
dc.journal.titleJournal of Mathematical Analysis and Applications
dc.language.isoeng
dc.publisherElsevier
dc.relation.projectIDPID2019-105979GB-I00
dc.relation.projectIDCEX2019-000904-S
dc.relation.projectID20205CEX001
dc.rights.accessRightsopen access
dc.subject.cdu517.553
dc.subject.keywordFinite Blaschke products
dc.subject.keywordCommutants
dc.subject.keywordReducing subspaces
dc.subject.keywordBergman spaces
dc.subject.ucmMatemáticas (Matemáticas)
dc.subject.ucmAnálisis matemático
dc.subject.unesco12 Matemáticas
dc.subject.unesco1202 Análisis y Análisis Funcional
dc.titleMultiplication by a finite Blaschke product on weighted Bergman spaces: Commutant and reducing subspaces
dc.typejournal article
dc.volume.number515
dspace.entity.typePublication
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