Reducing subspaces of Toeplitz operators induced by a class of non-analytic monomials over the unit ball. (English) Zbl 07904908
Summary: 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 \(M \geq 3\), 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.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
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