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

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2022

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Elsevier

##### Abstract

We 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.

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[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.