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Some properties of solutions to the Riccati equations in connection with Bergman spaces. (English) Zbl 1444.47024

The authors investigate the properties of solutions to operator Riccati equations in connection with Bergman spaces. Here, the Riccati equation of the form \(XAX+XB-CX-D=0\) is studied, where \(A, B, C, D\) are linear operators defined on Bergman spaces \(A_\alpha^2(\mathbb{D})\), the Hilbert space of all analytic function \(f\) on the unit disk with \[ \|f\|^2= \int_{\mathbb{D}} |f(z)|^2 (a+1)(1-|z|^2)^\alpha\, d A(z)<\infty, \] where \(dA\) is the area measure.
To solve Riccati equations in some concrete cases is an important problem in operator theory. This line has been attacked under various settings. For example, J.-J. Chen et al. [Front. Math. China 12, No. 4, 769–785 (2017; Zbl 1454.47020)] have characterized Riccati equations in connection with Dirichlet spaces. Some necessary conditions were given by M. T. Karaev [Monatsh. Math. 155, No. 2, 161–166 (2008; Zbl 1148.47013)] for the solvability of the Riccati equation by Toeplitz operators on Hardy space. The solvability of the Riccati operator equation on the Engliš algebra has also been studied [M. T. Karaev et al., Stud. Math. 232, No. 2, 113–141 (2016; Zbl 1372.47037)].
This paper characterizes some necessary conditions for the solvability of the Riccati equation on the set of all Toeplitz operators on the Bergman spaces \(A_\alpha^2(\mathbb{D})\), extending Karaev’s results on the Hardy space. It is worthy to mention that in studying the solvability of the Riccati equation, the authors obtain an interesting consequence that concerns the invariant subspaces of the Toeplitz operator.

MSC:

47A62 Equations involving linear operators, with operator unknowns
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A36 Bergman spaces of functions in several complex variables
47A15 Invariant subspaces of linear operators
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References:

[1] J. Chen, X. Wang, J. Xia, and G. Cao, “Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball”, Front. Math. China 12:4 (2017), 769-785. · Zbl 1454.47020
[2] M. Gürdal and F. Söhret, “Some results for Toeplitz operators on the Bergman space”, Appl. Math. Comput. 218:3 (2011), 789-793. · Zbl 1227.47017
[3] M. T. Karaev, “On the Riccati equations”, Monatsh. Math. 155:2 (2008), 161-166. · Zbl 1148.47013
[4] M. T. Karaev, M. Gürdal, and M. B. Huban, “Reproducing kernels, Engliš algebras and some applications”, Studia Math. 232:2 (2016), 113-141. · Zbl 1372.47037
[5] X. Zhao and D. Zheng, “Positivity of Toeplitz operators via Berezin transform”, J. Math. Anal. Appl. 416:2 (2014), 881-900. · Zbl 1325.47066
[6] K. · Zbl 1123.47001
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