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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.

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
Full Text: DOI
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