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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2511.02635 |
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| _version_ | 1866911248630677504 |
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| author | Pal, Avijit Paul, Bhaskar |
| author_facet | Pal, Avijit Paul, Bhaskar |
| contents | We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] = [\tilde{F}^*_j, \tilde{F}_{7-i}]$,$w(\tilde{F}^*_i + \tilde{F}_{7-i}z) \leqslant 1$ and these operators satisfy
\[(\tilde{F}^*_i + \tilde{F}_{7-i}z)Θ_{T_7}(z) = Θ_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] for $1 \leqslant i, j \leqslant 6$ for some $(F_1, \dots, F_6) \in \mathcal{B}(\mathcal{D}_{T_7})$ with $w(F^*_i + F_{7-i}z) \leqslant 1$ for $1 \leqslant i \leqslant 6$, then there exists a $Γ_{E(3; 3; 1, 1, 1)}$-contraction $(T_1, \dots, T_7)$ such that $F_1, \dots, F_6$ are the fundamental operators of $(T_1, \dots, T_7)$ and $\tilde{F}_1, \dots, \tilde{F}_6$ are the fundamental operators of $(T^*_1, \dots, T^*_7)$. We also prove similar type of result for pure $Γ_{E(3; 2; 1, 2)}$-contraction.
We explicitly construct a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, a $Γ_{E(3; 2; 1, 2)}$-unitary) starting from a $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, a $Γ_{E(3; 2; 1, 2)}$-contraction). Further, we develop functional models for general $Γ_{E(3; 3; 1, 1, 1)}$-isometries (respectively, $Γ_{E(3; 2; 1, 2)}$-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for $Γ_{E(3; 3; 1, 1, 1)}$-contractions (respectively, $Γ_{E(3; 2; 1, 2)}$-contractions). Finally, we present a Schaffer-type model for the $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation (respectively, the $Γ_{E(3; 2; 1, 2)}$-isometric dilation). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Admissible Fundamental Operators and Models for $Γ_{E(3; 3; 1, 1, 1)}$-contraction and $Γ_{E(3; 2; 1, 2)}$-contraction Pal, Avijit Paul, Bhaskar Functional Analysis We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] = [\tilde{F}^*_j, \tilde{F}_{7-i}]$,$w(\tilde{F}^*_i + \tilde{F}_{7-i}z) \leqslant 1$ and these operators satisfy \[(\tilde{F}^*_i + \tilde{F}_{7-i}z)Θ_{T_7}(z) = Θ_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] for $1 \leqslant i, j \leqslant 6$ for some $(F_1, \dots, F_6) \in \mathcal{B}(\mathcal{D}_{T_7})$ with $w(F^*_i + F_{7-i}z) \leqslant 1$ for $1 \leqslant i \leqslant 6$, then there exists a $Γ_{E(3; 3; 1, 1, 1)}$-contraction $(T_1, \dots, T_7)$ such that $F_1, \dots, F_6$ are the fundamental operators of $(T_1, \dots, T_7)$ and $\tilde{F}_1, \dots, \tilde{F}_6$ are the fundamental operators of $(T^*_1, \dots, T^*_7)$. We also prove similar type of result for pure $Γ_{E(3; 2; 1, 2)}$-contraction. We explicitly construct a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, a $Γ_{E(3; 2; 1, 2)}$-unitary) starting from a $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, a $Γ_{E(3; 2; 1, 2)}$-contraction). Further, we develop functional models for general $Γ_{E(3; 3; 1, 1, 1)}$-isometries (respectively, $Γ_{E(3; 2; 1, 2)}$-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for $Γ_{E(3; 3; 1, 1, 1)}$-contractions (respectively, $Γ_{E(3; 2; 1, 2)}$-contractions). Finally, we present a Schaffer-type model for the $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation (respectively, the $Γ_{E(3; 2; 1, 2)}$-isometric dilation). |
| title | Admissible Fundamental Operators and Models for $Γ_{E(3; 3; 1, 1, 1)}$-contraction and $Γ_{E(3; 2; 1, 2)}$-contraction |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.02635 |