Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866908420981915648 |
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| author | Bera, Santu |
| author_facet | Bera, Santu |
| contents | We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\mathcal D(μ_1,μ_2)$ on the unit bidisc $\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $μ_1,μ_2$ on $\overline{\mathbb D}.$ The multiplication operators $\mathscr M_{z_1}$ and $\mathscr M_{z_2}$ by the coordinate functions $z_1$ and $z_2,$ respectively, are bounded on $\mathcal D(μ_1,μ_2)$ and the set of polynomials is dense in $\mathcal D(μ_1,μ_2).$ We show that the commuting pair $\mathscr M_{z}=(\mathscr M_{z_1},\mathscr M_{z_2})$ is a cyclic analytic toral completely hyperexpansive $2$-tuple on $\mathcal D(μ_1,μ_2).$ Unlike the one variable case, not all cyclic analytic toral completely hyperexpansive pairs arise as multiplication $2$-tuple $\mathscr M_z$ on these spaces. In particular, we establish that a cyclic analytic toral completely hyperexpansive operator $2$-tuple $T=(T_1,T_2)$ satisfying $I-T^*_1 T_1-T^*_2T_2+T^*_1T^*_2T_1T_2=0$ and having a cyclic vector $f_0$ is unitarily equivalent to $\mathscr{M}_z$ on $\mathcal{D}(μ_1, μ_2)$ for some finite positive Borel measures $μ_1$ and $μ_2$ on $\overline{\mathbb{D}}$ if and only if $\ker T^*$, spanned by $f_0$, is a wandering subspace for $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators Bera, Santu Functional Analysis Primary 47A13, 32A10, 47B38, Secondary 31C05, 46E20 We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\mathcal D(μ_1,μ_2)$ on the unit bidisc $\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $μ_1,μ_2$ on $\overline{\mathbb D}.$ The multiplication operators $\mathscr M_{z_1}$ and $\mathscr M_{z_2}$ by the coordinate functions $z_1$ and $z_2,$ respectively, are bounded on $\mathcal D(μ_1,μ_2)$ and the set of polynomials is dense in $\mathcal D(μ_1,μ_2).$ We show that the commuting pair $\mathscr M_{z}=(\mathscr M_{z_1},\mathscr M_{z_2})$ is a cyclic analytic toral completely hyperexpansive $2$-tuple on $\mathcal D(μ_1,μ_2).$ Unlike the one variable case, not all cyclic analytic toral completely hyperexpansive pairs arise as multiplication $2$-tuple $\mathscr M_z$ on these spaces. In particular, we establish that a cyclic analytic toral completely hyperexpansive operator $2$-tuple $T=(T_1,T_2)$ satisfying $I-T^*_1 T_1-T^*_2T_2+T^*_1T^*_2T_1T_2=0$ and having a cyclic vector $f_0$ is unitarily equivalent to $\mathscr{M}_z$ on $\mathcal{D}(μ_1, μ_2)$ for some finite positive Borel measures $μ_1$ and $μ_2$ on $\overline{\mathbb{D}}$ if and only if $\ker T^*$, spanned by $f_0$, is a wandering subspace for $T$. |
| title | Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators |
| topic | Functional Analysis Primary 47A13, 32A10, 47B38, Secondary 31C05, 46E20 |
| url | https://arxiv.org/abs/2506.20143 |