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Some differential and integral equations with applications to Toeplitz operators. (English) Zbl 1023.45005
Let \(B_n\) be the open unit ball of the complex \(n\)-space \(\mathbb C^n\) and let \(H(B_n)\) denote the space of all holomorphic functions on \(B_n\). Let \(D\) be the radial derivative of \(f\in H(B_n)\) and \(P\) be the standard Bergman projection on \(B_n\).
The authors solve the differential equation \(fD(g)=gD(f)\) and system of integral equations \(fP(\overline z_kg)=gP(\overline z_kf)\), \(1\leq k\leq n\), to characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. Pluriharmonic symbols of normal Toeplitz operators and zero semi-commutators for certain classes of Toeplitz operators are characterized.

MSC:
45F05 Systems of nonsingular linear integral equations
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A10 Holomorphic functions of several complex variables
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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[1] S. Axler and ?. ?u?kovi?,Commuting Toeplitz Operators with Harmonic Symbols, Integr. Equ. Oper. Theory 14 (1991), 1-11 · Zbl 0733.47027
[2] B. R. Choe and Y. J. Lee,Pluriharmonic Symbols of Commuting Toeplitz Operators, Illinois J. of Math. 37 (1993), 424-436. · Zbl 0816.47024
[3] B. R. Choe and Y. J. Lee,Commuting Toeplitz Operators on the Harmonic Bergman Space, Michigan Math. J. 46 (1999) 163-174. · Zbl 0969.47023
[4] Y. J. Lee,Pluriharmonic Symbols of Commuting Toeplitz Type Operators on the Weighted Bergman Spaces, Canadian Math. Bull. 41 (2) (1998), 129-136. · Zbl 0920.47024
[5] W. Rudin, Function Theory in the Unit Ball of ? n , Springer-Verlag, Berlin, Heidelberg, New York, 1980. · Zbl 0495.32001
[6] D. Zheng,Commuting Toeplitz Operators with Pluriharmonic Symbols, Trans. of Amer. Math. Soc. 350 (1998), 1595-1618. · Zbl 0893.47015
[7] D. Zheng,Semi-commutators of Toeplitz Operators on the Bergman Space, Integr. Equ. Oper. Theory 25 (1996), 347-372 · Zbl 0864.47015
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