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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.26502 |
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| _version_ | 1866915586966028288 |
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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 $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$Γ_{E(3; 3; 1, 1, 1)}$-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\mathbf{T}$. Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators on $\mathcal{H}$ satisfying $S_i \tilde{S}_j = \tilde{S}_j S_i$ for $1 \leq i \leq 3$ and $1 \leq j \leq 2$. The tuple $\mathbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is called a \textit{$Γ_{E(3; 2; 1, 2)}$-contraction} if $Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\mathbf{S}$.
In this paper, we establish the existence and uniqueness of the fundamental operators associated with $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry.
We also construct a conditional dilation for a $Γ_{E(3; 3; 1, 1, 1)}$-contraction and a $Γ_{E(3; 2; 1, 2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction) admits a unique decomposition as a direct sum of a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, $Γ_{E(3; 2; 1, 2)}$-unitary) and a completely non-unitary $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Canonical Decompositions and Conditional Dilations of $Γ_{E(3; 3; 1, 1, 1)}$-Contraction and $Γ_{E(3; 2; 1, 2)}$-Contraction Keshari, Dinesh Kumar Pal, Avijit Paul, Bhaskar Functional Analysis A $7$-tuple of commuting bounded operators $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$Γ_{E(3; 3; 1, 1, 1)}$-contraction} if $Γ_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\mathbf{T}$. Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators on $\mathcal{H}$ satisfying $S_i \tilde{S}_j = \tilde{S}_j S_i$ for $1 \leq i \leq 3$ and $1 \leq j \leq 2$. The tuple $\mathbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is called a \textit{$Γ_{E(3; 2; 1, 2)}$-contraction} if $Γ_{E(3; 2; 1, 2)}$ is a spectral set for $\mathbf{S}$. In this paper, we establish the existence and uniqueness of the fundamental operators associated with $Γ_{E(3; 3; 1, 1, 1)}$-contractions and $Γ_{E(3; 2; 1, 2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure $Γ_{E(3; 3; 1, 1, 1)}$-isometry and a pure $Γ_{E(3; 2; 1, 2)}$-isometry. We also construct a conditional dilation for a $Γ_{E(3; 3; 1, 1, 1)}$-contraction and a $Γ_{E(3; 2; 1, 2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction) admits a unique decomposition as a direct sum of a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, $Γ_{E(3; 2; 1, 2)}$-unitary) and a completely non-unitary $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $Γ_{E(3; 2; 1, 2)}$-contraction). |
| title | Canonical Decompositions and Conditional Dilations of $Γ_{E(3; 3; 1, 1, 1)}$-Contraction and $Γ_{E(3; 2; 1, 2)}$-Contraction |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.26502 |