Abstract
Last chapter mainly concerns with reducing subspace problem of multiplication operators M B induced by finite Blaschke products B. This chapter still focuses on the same theme, whereas the symbol B is replaced with a thin Blaschke product. In Chap. 3 it was shown that the geometric property of this symbol B is a key to the study of the abelian property of \(\mathcal{V}^{{\ast}}(B)\). However, the geometry of thin Blaschke products is far more complicated than that of finite Blaschke products.
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Bibliography
M. Abrahamse, Analytic Toeplitz operators with automorphic symbol. Proc. Am. Math. Soc. 52, 297–302 (1975)
J. Ball, Hardy space expectation operators and reducing subspaces. Proc. Am. Math. Soc. 47, 351–357 (1975)
I. Chalendar, E. Fricain, D. Timotin, Functional models and asymptotically orthonormal sequences. Ann. Inst. Fourier (Grenoble) 53, 1527–1549 (2003)
J. Conway, Subnormal Operators. Research Notes in Mathematics, vol. 51 (Pitman Advanced Publishing Program, Boston, 1981)
C. Cowen, Finite Blaschke products as composition of other finite Blaschke products. arXiv: math.CV/ 1207.4010v1
R. Douglas, M. Putinar, K. Wang, Reducing subspaces for analytic multipliers of the Bergman space. J. Funct. Anal. 263, 1744–1765 (2012) arXiv: math.FA/ 1110.4920v1
R. Douglas, S. Sun, D. Zheng, Multiplication operators on the Bergman space via analytic continuation. Adv. Math. 226, 541–583 (2011)
J. Deddens, T. Wong, The commutant of analytic Toeplitz operators. Trans. Am. Math. Soc. 184, 261–273 (1973)
J. Garnett, Bounded Analytic Functions (Academic, New York, 1981)
K. Guo, 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, 1219–1255 (2011)
K. Guo, H. Huang, Geometric constructions of thin Blaschke products and reducing subspace problem. Proc. Lond. Math. Soc. 109, 1050–1091 (2014)
P. Gorkin, R. Mortini, Asymptotic interpolating sequences in uniform algebras. J. Lond. Math. Soc. 67, 481–498 (2003)
K. Guo, S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space via the Hardy space of the bidisk. J. Reine Angew. Math. 629, 129–168 (2009)
A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)
K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, 1962)
J. Hu, S. Sun, X. Xu, D. Yu, Reducing subspace of analytic Toeplitz operators on the Bergman space. Integr. Equ. Oper. Theory 49, 387–395 (2004)
J. Milnor, Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160 (Princeton University Press, Princeton, 2006)
Z. Nehari, Conformal Mapping (McGraw-Hill, New York, 1952)
N. Nikolski, Treatise on the Shift Operator. Grundlehren der mathematischen Wissenschafte, vol. 273 (Springer, Berlin, 1986)
E. Nordgren, Reducing subspaces of analytic Toeplitz operators. Duke Math. J. 34, 175–181 (1967)
J. Ritt, Prime and composite polynomials. Trans. Am. Math. Soc. 23, 51–66 (1922)
J. Ritt, Permutable rational functions. Trans. Am. Math. Soc. 25, 399–448 (1923)
W. Rudin, A generalization of a theorem of Frostman. Math. Scand. 21, 136–143 (1967)
M. Stessin, K. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators. Can. J. Math. 55, 379–400 (2003)
S. Sun, D. Zheng, C. Zhong, Classification of reducing subspaces of a class of multiplication operators via the Hardy space of the bidisk. Can. J. Math. 62, 415–438 (2010)
J. Thomson, The commutant of a class of analytic Toeplitz operators. Am. J. Math. 99, 522–529 (1977)
J. Thomson, The commutant of a class of analytic Toeplitz operators II. Indiana Univ. Math. J. 25, 793–800 (1976)
J. Thomson, The commutant of certain analytic Toeplitz operators. Proc. Am. Math. Soc. 54, 165–169 (1976)
J. Walsh, On the location of the roots of the jacobian of two binary forms, and of the derivative of a rational function. Trans. Am. Math. Soc. 19, 291–298 (1918)
J. Walsh, The Location of Critical Points, vol. 34 (American Mathematical Society Colloquium Publications, Rhode Island, 1950)
K. Zhu, Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. 62, 553–568 (2000)
K. Zhu, Irreducible multiplication operators on spaces of analytic functions. J. Oper. Theory 51, 377–385 (2004)
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Guo, K., Huang, H. (2015). Reducing Subspaces Associated with Thin Blaschke Products. In: Multiplication Operators on the Bergman Space. Lecture Notes in Mathematics, vol 2145. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46845-6_5
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