Abstract
In this paper, we describe the minimal reducing subspaces of Toeplitz operators induced by non-analytic monomials on the weighted Bergman spaces and Dirichlet spaces over the unit ball \({\mathbb{B}_2}\). It is proved that each minimal reducing subspace M is finite dimensional, and if dim M ≥ 3, then M is induced by a monomial. Furthermore, the structure of commutant algebra \(\nu ({T_{\overline w {N_z}N}}): = {\{ M_{{w^N}}^ * {M_{{z^N}}},M_{{z^N}}^ * {M_{{w^N}}}\} ^\prime }\) is determined by N and the two dimensional minimal reducing subspaces of \({T_{\overline w {N_z}N}}\). We also give some interesting examples.
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Acknowledgements
The authors would like to thank Caixing Gu for his many helpful suggestions and comments. We also thank the referees for their comments.
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Supported by NSFC (Grant Nos. 12031002, 12371134) and SDNSFC (Grant Nos. ZR2021MA015, ZR2020 MA009)
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Shi, Y.Y., Zhang, B., Tang, X. et al. Reducing Subspaces of Toeplitz Operators Induced by a Class of Non-analytic Monomials over the Unit Ball. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2709-x
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DOI: https://doi.org/10.1007/s10114-024-2709-x