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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.16541 |
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| _version_ | 1866913497708756992 |
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| author | Bhattacharjee, Monojit Gupta, Rajeev Venugopal, Vidhya |
| author_facet | Bhattacharjee, Monojit Gupta, Rajeev Venugopal, Vidhya |
| contents | In this article, we define Dirichlet-type space $\mathcal{D}^{2}(\boldsymbolμ)$ over the bidisc $\mathbb D^2$ for any measure $\boldsymbolμ\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ We show that the set of polynomials is dense in $\mathcal{D}^{2}(\boldsymbolμ)$ and the pair $(M_{z_1}, M_{z_2})$ of multiplication operator by co-ordinate functions on $\mathcal{D}^{2}(\boldsymbolμ)$ is a pair of commuting $2$-isometries. Moreover, the pair $(M_{z_1}, M_{z_2})$ is a left-inverse commuting pair in the following sense: $L_{M_{z_i}} M_{z_j}=M_{z_j}L_{M_{z_i}}$ for $1\leqslant i\neq j\leqslant n,$ where $L_{M_{z_i}}$ is the left inverse of $M_{z_i}$ with $\ker L_{M_{z_i}} =\ker M_{z_i}^*$, $1\leqslant i \leqslant n$. Furthermore, it turns out that, for the class of left-inverse commuting tuple $\boldsymbol T=(T_1, \ldots, T_n)$ acting on a Hilbert space $\mathcal{H}$, the joint wandering subspace property is equivalent to the individual wandering subspace property. As an application of this, the article shows that the class of left-inverse commuting pair with certain splitting property is modelled by the pair of multiplication by co-ordinate functions $(M_{z_1}, M_{z_2})$ on $\mathcal{D}^{2}(\boldsymbolμ)$ for some $\boldsymbolμ\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16541 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet type spaces in the unit bidisc and Wandering Subspace Property for operator tuples Bhattacharjee, Monojit Gupta, Rajeev Venugopal, Vidhya Functional Analysis In this article, we define Dirichlet-type space $\mathcal{D}^{2}(\boldsymbolμ)$ over the bidisc $\mathbb D^2$ for any measure $\boldsymbolμ\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ We show that the set of polynomials is dense in $\mathcal{D}^{2}(\boldsymbolμ)$ and the pair $(M_{z_1}, M_{z_2})$ of multiplication operator by co-ordinate functions on $\mathcal{D}^{2}(\boldsymbolμ)$ is a pair of commuting $2$-isometries. Moreover, the pair $(M_{z_1}, M_{z_2})$ is a left-inverse commuting pair in the following sense: $L_{M_{z_i}} M_{z_j}=M_{z_j}L_{M_{z_i}}$ for $1\leqslant i\neq j\leqslant n,$ where $L_{M_{z_i}}$ is the left inverse of $M_{z_i}$ with $\ker L_{M_{z_i}} =\ker M_{z_i}^*$, $1\leqslant i \leqslant n$. Furthermore, it turns out that, for the class of left-inverse commuting tuple $\boldsymbol T=(T_1, \ldots, T_n)$ acting on a Hilbert space $\mathcal{H}$, the joint wandering subspace property is equivalent to the individual wandering subspace property. As an application of this, the article shows that the class of left-inverse commuting pair with certain splitting property is modelled by the pair of multiplication by co-ordinate functions $(M_{z_1}, M_{z_2})$ on $\mathcal{D}^{2}(\boldsymbolμ)$ for some $\boldsymbolμ\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ |
| title | Dirichlet type spaces in the unit bidisc and Wandering Subspace Property for operator tuples |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2406.16541 |