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A theorem of Brown-Halmos type for Bergman space Toeplitz operators. (English) Zbl 0996.47037

From author’s abstract: We study the analogous of the Brown-Halmos theorem for Toeplitz operators on the Bergman space. We show that for \(f\) and \(g\) harmonic, \(T_f T_g= T_h\) only in the trivial case, provided that \(h\) is of class \(C^2\) with the invariant Laplacian bounded. Here the trivial cases are \(\overline f\) or \(g\) holomorphic. From this we conclude that the zero-product problem for harmonic symbols has only the trivial solution. Finally, we provide examples that show the Brown-Halmos theorem fails for general symbols, even for symbols continuous up to the boundary.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A60 Functional calculus for linear operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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References:

[1] S. Axler, Berman Spaces and Their Operators, Pitman Research Notes in Mathematics, Vol. 171, pp. 1-50, Longman, Harlow.; S. Axler, Berman Spaces and Their Operators, Pitman Research Notes in Mathematics, Vol. 171, pp. 1-50, Longman, Harlow.
[2] Brown, A.; Halmos, P., Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213, 89-102 (1964) · Zbl 0116.32501
[3] Engliš, M., Berezin transform and the Laplace-Beltrami operator, St. Petersburg Math. J., 7, 633-647 (1996) · Zbl 0859.58033
[4] Zheng, D., Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal., 83, 98-120 (1989) · Zbl 0678.47026
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