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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2510.25666 |
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| _version_ | 1866908619179556864 |
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| author | Keshari, Dinesh Kumar Pal, Avijit Paul, Bhaskar |
| author_facet | Keshari, Dinesh Kumar Pal, Avijit Paul, Bhaskar |
| contents | A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$Γ_{E(3; 3; 1, 1, 1)} $-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $ Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators defined on a Hilbert space $\mathcal{H}$ with $S_i\tilde{S}_j = \tilde{S}_jS_i$ for $1 \leqslant i \leqslant 3$ and $1 \leqslant j \leqslant 2$. We say that $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is a $Γ_{E(3; 2; 1, 2)} $-contraction if $ Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. We derive various properties of $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for $Γ_{E(3; 3; 1, 1,1 )}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. We explore the structure of $Γ_{E(3; 3; 1, 1, 1)}$-unitaries and $Γ_{E(3; 2; 1, 2)}$-unitaries and elaborate on the relationship between them. We also study various properties of $Γ_{E(3; 3; 1, 1, 1)}$-isometries and $Γ_{E(3; 2; 1, 2)}$-isometries. We discuss the Wold Decomposition for a $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a $Γ_{E(3; 2; 1, 2)}$-isometry. We further outline the structure theorem for a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Operators on Hilbert Space having $Γ_{E(3; 3; 1, 1, 1)}$ and $Γ_{E(3; 2; 1, 2)}$ as Spectral Sets Keshari, Dinesh Kumar Pal, Avijit Paul, Bhaskar Functional Analysis A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$Γ_{E(3; 3; 1, 1, 1)} $-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $ Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators defined on a Hilbert space $\mathcal{H}$ with $S_i\tilde{S}_j = \tilde{S}_jS_i$ for $1 \leqslant i \leqslant 3$ and $1 \leqslant j \leqslant 2$. We say that $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is a $Γ_{E(3; 2; 1, 2)} $-contraction if $ Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. We derive various properties of $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for $Γ_{E(3; 3; 1, 1,1 )}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. We explore the structure of $Γ_{E(3; 3; 1, 1, 1)}$-unitaries and $Γ_{E(3; 2; 1, 2)}$-unitaries and elaborate on the relationship between them. We also study various properties of $Γ_{E(3; 3; 1, 1, 1)}$-isometries and $Γ_{E(3; 2; 1, 2)}$-isometries. We discuss the Wold Decomposition for a $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a $Γ_{E(3; 2; 1, 2)}$-isometry. We further outline the structure theorem for a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry. |
| title | Operators on Hilbert Space having $Γ_{E(3; 3; 1, 1, 1)}$ and $Γ_{E(3; 2; 1, 2)}$ as Spectral Sets |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.25666 |