Reducing subspaces for \(T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}\) on weighted Hardy space over bidisk. (English) Zbl 1498.47070
Summary: In this paper, we characterize the reducing subspaces for Toeplitz operator \(T = M_{z^k} + M^*_{z^l}\), where \(M_{z^k}\), \(M_{z^l}\) are the multiplication operators on weighted Hardy space \(\mathcal{H}_\omega^2(\mathbb{D}^2)\), \(k =(k_1, k_2)\), \(l=(l_1,l_2)\), \(k\neq l\) and \(k_i, l_i\) are positive integers for \(i=1, 2\). It is proved that the reducing subspace for \(T\) generated by \(z^m\) is minimal under proper assumptions on \(\omega\). The Bergman space and weighted Dirichlet spaces \(\mathcal{D}_\delta(\mathbb{D}^2)\) (\(\delta>0\)) are weighted Hardy spaces which satisfy these assumptions. As an application, we describe the reducing subspaces for \(T_{z^k+\bar{z}^l}\) on \(\mathcal{D}_\delta(\mathbb{D}^2)\) (\(\delta > 0\)), which generalizes the results on Bergman space over bidisk.
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 32A35 | \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables |
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