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- W4387156613 abstract "A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a textit{$mathbb{P}$-contraction} if the closed pentablock $overline{mathbb P}$ is a spectral set for $(A,S,P)$, where [ mathbb{P}:=left{(a_{21}, mbox{tr}(A_0), mbox{det}(A_0)) : A_0=[a_{ij}]_{2 times 2} ; ; & ;; |A_0| <1 right} subseteq mathbb{C}^3. ] We find several characterizations for the $mathbb P$-unitaries and $mathbb P$-isometries. We show that every $mathbb P$-isometry admits a Wold type decomposition that splits it into a direct sum of a $mathbb P$-unitary and a pure $mathbb P$-isometry. We also show that every $mathbb P$-contraction $(A,S,P)$ possesses a canonical decomposition that orthogonally decomposes $(A,S,P)$ into a $mathbb P$-unitary and a completely non-unitary $mathbb P$-contraction. We find a necessary and sufficient condition such that a $mathbb P$-contraction $(A, S, P)$ dilates to a $mathbb P$-isometry $(X, T, V)$ with $V$ being the minimal isometric dilation of $P$. Then we show an explicit construction of such a conditional dilation. A commuting tuple of Hilbert space operators $(T_1, dots , T_n)$ having the closed unit ball $overline{mathbb B}_n$ as a spectral set is called a $mathbb B_n$-textit{contraction} and a commuting pair $(S,P)$ having the closed symmetrized bidisc $overline{mathbb G_2} ,(=Gamma)$ as a spectral set is called a $Gamma$-textit{contraction}. We characterize isometries and unitaries associated with $mathbb B_n$. Then we present an analogous canonical decomposition for a $mathbb B_n$-contraction. We show interplay between operator theory on the three domains $mathbb P, mathbb B_2$ and $mathbb G_2$." @default.
- W4387156613 created "2023-09-30" @default.
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- W4387156613 date "2023-09-26" @default.
- W4387156613 modified "2023-09-30" @default.
- W4387156613 title "Operators associated with the pentablock and their relations with biball and symmetrized bidisc" @default.
- W4387156613 doi "https://doi.org/10.48550/arxiv.2309.15080" @default.
- W4387156613 hasPublicationYear "2023" @default.
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