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Commuting Toeplitz operators on the harmonic Bergman space. (English) Zbl 0969.47023
The authors prove the following results:
(1) Let \(u,v\in b^2\), and suppose that \(T_u\) and \(T_v\) commute on \(b^2\). If \(\partial u\) and \(\partial v\) are both not identically zero, then there exists a constant \(\alpha\) such that \(\partial v=\alpha(\partial u)\).
(2) Let \(f,g\in L^2_a\) be nonconstant functions. Then \(T_fT_g= T_gT_f\) on \(b^2\) if and only if \(g=\alpha f+\beta\) for some constants \(\alpha\) and \(\beta\).
(3) Let \(u\in b^2\). Then \(T_u\) is normal on \(b^2\) if and only if \(u(D)\) is contained in a straight line. In particular, for \(f\in L^2_a\), \(T_f\) is normal on \(b^2\) if and only if \(f\) is constant.
(4) Let \(u,v\in b^2\) be nonconstant functions and suppose one of them is a polynomial. Then \(T_u\) and \(T_v\) commute on \(b^2\) if and only if \(v=\alpha u+\beta\) for some constants \(\alpha\), \(\beta\).
Reviewer: C.Lai (Zhangzhou)

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H05 Spaces of bounded analytic functions of one complex variable
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