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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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