The reducibility of compressed shifts on a class of quotient modules over the bidisk. (English) Zbl 1477.47027

It is shown that, for the rational inner function \(\theta(z,w)=\frac{zw+\bar{b}w+\bar{c}z+\bar{d}}{1+bz+cw+dzw}\), the compressed shift \(S_z=P_\theta T_z|_{\mathcal{K}_\theta}\) is reducible on the quotient module \(\mathcal{K}_\theta=H^2\ominus \theta H^2\) over the bidisk if and only if \(\theta\) is the product of two one-variable inner functions.


47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A15 Invariant subspaces of linear operators
32A10 Holomorphic functions of several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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[1] J. Agler, “On the representation of certain holomorphic functions defined on a polydisc” in Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl. 48, Birkhäuser, Basel, 1990, 47-66. · Zbl 0733.32002
[2] J. A. Ball, C. Sadosky, and V. Vinnikov, Scattering systems with several evolutions and multidimensional input/state/output systems, Integral Equations Operator Theory 52 (2005), no. 3, 323-393. · Zbl 1092.47006 · doi:10.1007/s00020-005-1351-y
[3] K. Bickel and P. Gorkin, Compression of the shift on the bidisk and their numerical ranges, J. Operator Theory 79 (2018), no. 1, 225-265. · Zbl 1399.47029
[4] K. Bickel and C. Liaw, Properties of Beurling-type submodules via Agler decompositions, J. Funct. Anal. 272 (2017), no. 1, 83-111. · Zbl 1417.47004 · doi:10.1016/j.jfa.2016.10.007
[5] R. G. Douglas and C. Foias, “On the structure of the square of a \(C_0(1)\) operator” in Modern Operator Theory and Applications, Oper. Theory Adv. Appl. 170, Birkhäuser, Basel, 2007, 75-84. · Zbl 1119.47010
[6] R. G. Douglas, M. Putinar, and K. Wang, Reducing subspaces for analytic multipliers of the Bergman space, J. Funct. Anal. 263 (2012), no. 6, 1744-1765. · Zbl 1275.47071 · doi:10.1016/j.jfa.2012.06.008
[7] J. B. Garnett, Bounded Analytic Functions, Grad. Texts in Math. 236, Springer, New York, 2007.
[8] K. Guo and H. Huang, Multiplication Operators on the Bergman Space, Lecture Notes in Math. 2145, Springer, Heidelberg, 2015. · Zbl 1321.47002
[9] K. Guo and K. Wang, Beurling type quotient modules over the bidisk and boundary representations, J. Funct. Anal. 257 (2009), no. 10, 3218-3238. · Zbl 1178.47002 · doi:10.1016/j.jfa.2009.06.031
[10] G. Knese, Integrability and regularity of rational functions, Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 1261-1306. · Zbl 1341.32008 · doi:10.1112/plms/pdv061
[11] Y. Li, Y. Yang, and Y. Lu, Reducibility and unitary equivalence for a class of truncated Toeplitz operators on model spaces, New York J. Math. 24 (2018), 929-946. · Zbl 06982022
[12] W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969. · Zbl 0177.34101
[13] K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. (2) 62 (2000), no. 2, 553-568. · Zbl 1158.47309 · doi:10.1112/S0024610700001198
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