Abstract
Let \({\mathfrak{D}} \) be the Dirichlet space on the unit disc \({\mathbb{D}}\) and B(z) be the Blaschke product with n zero, and we prove that multiplication operator \(M_B\) on \({\mathfrak{D}}\) is similar to \(\bigoplus _{1}^{n}M_{z}\) on \(\bigoplus _{1}^{n}{\mathfrak{D}}\) by a crucial decomposition of \({\mathfrak{D}}.\) Two applications of our similarity theorem are presented. First, we characterize the intersection of the commutant of multiplication operator \(M_B\) on the Dirichlet space setting from the techniques in operator theory combined with matrix manipulations. Second, we give a necessary and sufficient condition for the similarity of two multiplication operators \(M_{f_1}\) and \(M_{f_2}\) on \({\mathfrak{D}},\) where \(f_{1}\) and \(f_{2}\) are analytic functions on the closure of the unit disc \({\mathbb{D}}.\)
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References
Axler, S.: The Bergman space, the Bloch spaces and commutators of multiplication operators. Duke Math. J. 53, 315–332 (1986)
Ball, J.: Hardy space expectation operators and reducing subspaces. Proc. Am. Math. Soc. 47(2), 351–357 (1975)
Brown, L., Shields, A.: Cyclic vectors in the Dirichlet space. Trans. Am. Math. Soc. 285(1), 269–303 (1984)
Cao, G.: Fredholm properties of Toeplitz operators on Dirichlet spaces. Pac. J. Math. 188, 209–223 (1999)
Cowen, C.: The commutant of an analytic Toeplitz operator. Trans. Am. Math. Soc. 239, 1–31 (1978)
Cowen, C.: On equivalence of Toeplitz operators. J. Oper. Theory 7, 167–172 (1982)
Deddens, J., Wong, T.: The commutant of analytic Toeplitz operator. Trans. Am. Math. Soc. 184, 261–273 (1973)
Douglas, R.: Operator theory and complex geometry. Extr. Math. 24, 135–165 (2009)
El-Fallah, O., Elmadani, Y., Kellay, K.: Cyclicity and invariant subspaces in Dirichlet spaces. J. Funct. Anal. 270(9), 3262–3279 (2016)
El-Fallah, O., Kellay, K., Mashreghi, J., Ransford, T.: A Primer on the Dirichlet Space. Cambridge Tracts in Mathematics, vol. 203. Cambridge University Press, Cambridge (2014)
Gallardo-Gutierrez, E., Partington, J.: Multiplication by a finite Blaschke product on weighted Bergman spaces: commutant and reducing subspaces. J. Math. Anal. Appl. 515, 126383 (2022)
Ghosh, G.: Multiplication operators on the Bergman space by proper holomorphic map**s. J. Math. Anal. Appl. 510, 126026 (2022)
Guo, K., Huang, H.: On multiplication operators of the Bergman space: similarity, unitary equivalence and reducing subspaces. J. Oper. Theory 65(2), 355–378 (2011)
Guo, K., Huang, H.: Multiplication Operators on the Bergman Space. Lecture Notes in Mathematics, vol. 2145. Springer, Berlin (2015)
Jiang, C., Li, Y.: The commutant and similarity invariant of analytic Toeplitz operators on Bergman space. Sci. China Ser. A: Math. 50(5), 651–664 (2007)
Jiang, C., Zheng, D.: Similarity of analytic Toeplitz operators on the Bergman spaces. J. Funct. Anal. 258, 2961–2982 (2010)
Li, L., Nakada, Y., Nestor, D., Shang, W., Weir, R.: Normal weighted composition operators on weighted Dirichlet spaces. J. Math. Anal. Appl. 423, 758–769 (2015)
Li, Y., Zhang, Z., Zheng, D.: Kernels and spectrum of Toeplitz operators on the Dirichlet space. J. Math. Anal. Appl. 472(1), 894–919 (2019)
Luecking, D.: Inequalities on Bergman spaces. Ill. J. Math. 25, 1–11 (1981)
Richter, S., Sundberg, C.: Multipliers and invariant subspaces in the Dirichlet space. J. Oper. Theory 28(1), 167–186 (1992)
Rochberg, R., Wu, Z.: Toeplitz operators on Dirichlet spaces. Integral Equ. Oper. Theory 15, 325–342 (1992)
Rordam, M., Larsen, F., Laustsen, N.: An Introduction to \(K\)-Theory for \(C^{\ast }\)-Algebras, vol. 49. Cambridge University Press, Cambridge (2000)
Sarason, D.: Weak-star generators of \(H^{\infty }\). Pac. J. Math. 17, 519–528 (1966)
Shapiro, J.: Composition Operators and Classical Function Theory. Springer, New York (1993)
Stegenga, D.: Multipliers of the Dirichlet space. Ill. J. Math. 24(1), 113–139 (1980)
Stessin, M., Zhu, K.: Generalized factorization in Hardy spaces and the commutant of Toeplitz operators. Can. J. Math. 55(2), 379–400 (2003)
Thomson, J.: Intersection of commutants of analytic Toeplitz operator. Proc. Am. Math. Soc. 52, 305–310 (1975)
Walsh, J.: The Location of Critical Points of Analytic and Harmonic Function, vol. 34. American Mathematical Society Colloquium Publications, Providence (1950)
Yu, T.: Toeplitz operators on the Dirichlet space. Integral Equ. Oper. Theory 67, 163–170 (2010)
Zhu, K.: Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. 62(2), 553–568 (2000)
Acknowledgements
The authors would like to thank the anonymous referee for a meticulous reading and many valuable suggestions to improve this paper. This work is partially supported by NSFC (12171138, 11801572, 12171484) and Central South University Innovation-Driven Research Programme (2023CXQD032).
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Communicated by Gelu Popescu.
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Li, Y., Ma, P. On similarity and commutant of a class of multiplication operators on the Dirichlet space. Banach J. Math. Anal. 17, 9 (2023). https://doi.org/10.1007/s43037-022-00234-1
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DOI: https://doi.org/10.1007/s43037-022-00234-1