Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.08565 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916214702342144 |
|---|---|
| author | Dahya, Raj |
| author_facet | Dahya, Raj |
| contents | We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of contractive $\mathcal{C}_{0}$-semigroups on $L^{2}(\prod_{i \in \mathcal{I}}\mathbb{T}) \otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for $d \in \mathbb{N}$ with $d \geq 3$ the existence of commuting families $\{T_{i}\}_{i=1}^{d}$ of contractive $\mathcal{C}_{0}$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive $\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces for each $d \geq 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, \textit{viz.} that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of $\mathcal{C}_{0}$-semigroups, which extends results of Eisner (2009--10). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08565 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Interpolation and non-dilatable families of $\mathcal{C}_{0}$-semigroups Dahya, Raj Functional Analysis 47A20, 47D06 We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of contractive $\mathcal{C}_{0}$-semigroups on $L^{2}(\prod_{i \in \mathcal{I}}\mathbb{T}) \otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for $d \in \mathbb{N}$ with $d \geq 3$ the existence of commuting families $\{T_{i}\}_{i=1}^{d}$ of contractive $\mathcal{C}_{0}$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive $\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces for each $d \geq 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, \textit{viz.} that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of $\mathcal{C}_{0}$-semigroups, which extends results of Eisner (2009--10). |
| title | Interpolation and non-dilatable families of $\mathcal{C}_{0}$-semigroups |
| topic | Functional Analysis 47A20, 47D06 |
| url | https://arxiv.org/abs/2307.08565 |