Commuting families of polygonal type operators on Hilbert space
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2024
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| author | Merdy, Christian Le Reshmi, M. N. |
| author_facet | Merdy, Christian Le Reshmi, M. N. |
| contents | Let $T\colon H\to H$ be a bounded operator on Hilbert space. We say that $T$ has a polygonal type if there exists an open convex polygon $Δ\subset {\mathbb D}$, with $\overlineΔ\cap{\mathbb T}\neq\emptyset$, such that the spectrum $σ(T)$ is included in $\overlineΔ$ and the resolvent $R(z,T)$ satisfies an estimate $\Vert R(z,T)\Vert \lesssim \max\{\vert z-ξ\vert^{-1}\, :\, ξ\in \overlineΔ\cap{\mathbb T}\}$ for $z\in\overline{\mathbb D}^c$. The class of polygonal type operators (which goes back to De Laubenfels and Franks-McIntosh) contains the class of Ritt operators. Let $T_1,\ldots,T_d$ be commuting operators on $H$, with $d\geq 3$. We prove functional calculus properties of the $d$-tuple $(T_1,\ldots,T_d)$ under various assumptions involving poygonal type. The main ones are the following. (1) If the $T_k$ are contractions for all $k=1,\ldots,d$ and if $T_1,\ldots,T_{d-2}$ have a polygonal type, then $(T_1,\ldots,T_d)$ satisfies a generalized von Neumann inequality $\Vert ϕ(T_1,\ldots,T_d)\Vert \leq C\Vertϕ\Vert_{\infty,{\mathbb D}^d}$ for polynomials $ϕ$ in $d$ variables; (2) If $T_k$ is polynomially bounded with a polygonal type for all $k=1,\ldots,d$, then there exists an invertible operator $S\colon H\to H$ such that $\Vert S^{-1}T_kS\Vert \leq 1$ for all $k=1,\ldots,d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_12321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commuting families of polygonal type operators on Hilbert space Merdy, Christian Le Reshmi, M. N. Functional Analysis 47A60, 47A20, 47A13 Let $T\colon H\to H$ be a bounded operator on Hilbert space. We say that $T$ has a polygonal type if there exists an open convex polygon $Δ\subset {\mathbb D}$, with $\overlineΔ\cap{\mathbb T}\neq\emptyset$, such that the spectrum $σ(T)$ is included in $\overlineΔ$ and the resolvent $R(z,T)$ satisfies an estimate $\Vert R(z,T)\Vert \lesssim \max\{\vert z-ξ\vert^{-1}\, :\, ξ\in \overlineΔ\cap{\mathbb T}\}$ for $z\in\overline{\mathbb D}^c$. The class of polygonal type operators (which goes back to De Laubenfels and Franks-McIntosh) contains the class of Ritt operators. Let $T_1,\ldots,T_d$ be commuting operators on $H$, with $d\geq 3$. We prove functional calculus properties of the $d$-tuple $(T_1,\ldots,T_d)$ under various assumptions involving poygonal type. The main ones are the following. (1) If the $T_k$ are contractions for all $k=1,\ldots,d$ and if $T_1,\ldots,T_{d-2}$ have a polygonal type, then $(T_1,\ldots,T_d)$ satisfies a generalized von Neumann inequality $\Vert ϕ(T_1,\ldots,T_d)\Vert \leq C\Vertϕ\Vert_{\infty,{\mathbb D}^d}$ for polynomials $ϕ$ in $d$ variables; (2) If $T_k$ is polynomially bounded with a polygonal type for all $k=1,\ldots,d$, then there exists an invertible operator $S\colon H\to H$ such that $\Vert S^{-1}T_kS\Vert \leq 1$ for all $k=1,\ldots,d$. |
| title | Commuting families of polygonal type operators on Hilbert space |
| topic | Functional Analysis 47A60, 47A20, 47A13 |
| url | https://arxiv.org/abs/2407.12321 |