Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball

Abstract

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX+ XB−CX= Drelated to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

Keywords

Riccati equation / Toeplitz operator / Dirichlet space / unit ball

References

[1] BergerC A, CoburnL A, ZhuK H. Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus. Amer J Math, 1988, 110(5): 921–953

[2] CaoG F. Toeplitz operators on Dirichlet spaces. Chinese J Contemp Math, 2000, 21(3): 283–298 (in Chinese)

[3] CaoG F, ZhuL T. The compactness of Toeplitz operators on Dirichlet spaces. Acta Math Sinica (Chin Ser), 2001, 44(2): 241–248 (in Chinese)

[4] GürdalM, SöhretF. Some results for Toeplitz operators on the Bergman space. Appl Math Comput, 2011, 218(3): 789–793

[5] KaraevM T. On the Riccati equations. Monatsh Math, 2008, 155(2): 161–166

[6] KaraevM T, GürdalM. On the Berezin symbols and Toeplitz operators. Extracta Math, 2010, 25(1): 83–102

[7] LiS X, HuJ Y. Compact operators on Bergman spaces of the unit ball. Acta Math Sinica(Chin Ser), 2004, 47(5): 837–844 (in Chinese)

[8] LuY F, SunS H. Toeplitz operators on Dirichlet spaces. Acta Math Sin (Engl Ser), 2001, 17(4): 643–648

[9] NordgrenE, RosenthalP. Boundary values of Berezin symbols. Oper Theory Adv Appl, 1994, 73: 362–368

[10] WangX F, XiaJ, CaoG F. Some properties of Toeplitz operators on Dirichlet space with respect to Berezin type transform. Sci Sin (Math), 2012, 42(11): 1159–1170

[11] XiaJ, WangX F, CaoG F. Compact operators on Dirichlet space. Acta Math Sci Ser A Chin Ed, 2009, 29: 1196–1205 (in Chinese)

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2017 Higher Education Press and Springer-Verlag Berlin Heidelberg