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Commutants of analytic Toeplitz operators on the harmonic Bergman space. (English) Zbl 1088.47016
The analytic Bergman space $$L^2_a$$ is the closed subspace of functions in $$L^2$$ of the open unit disk that are analytic, while the harmonic Bergman space $$b^2$$ is the closed subspace of functions in $$L^2$$ that are harmonic on the disk. The orthogonal projections of $$L^2$$ onto $$L^2_a$$ and $$b^2$$ are denoted by $$P$$ and $$Q$$, respectively. For $$u\in L^\infty$$ of the disk, the Toeplitz operator $$T_u$$ on $$b^2$$ is defined by $$T_uf= Q(uf)$$.
In their previous work [Mich. Math. J. 46, No. 1, 163–174 (1999; Zbl 0969.47023)], the authors proved that two analytic Toeplitz operators on $$b^2$$ commute only when their symbols and 1 are linearly dependent. The main result of this paper is as follows.
Theorem. Let $$f\in H^\infty$$ and $$u\in L^\infty$$. Assume that $$f$$ is nonconstant and $$P\overline u$$ is not a cyclic vector for the adjoint of the shift operator on $$L^2_a$$. Then $$T_uT_f= T_fT_u$$ if and only if $$u$$ is a linear combination of $$f$$ and 1.
Whether the noncyclicity hypothesis can be removed remains open. The authors also obtain the analogous result for semi-commutants.

##### MSC:
 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
##### Keywords:
commutant; Toeplitz operator; harmonic Bergman space
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