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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.

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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