Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.08694 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910496178831360 |
|---|---|
| author | Kojin, Kenta |
| author_facet | Kojin, Kenta |
| contents | We study a Schwarz-Pick type inequality for the Schur-Agler class $SA(B_δ)$. In our operator theoretical approach, von Neumann's inequality for a class of generic tuples of $2\times 2$ matrices plays an important role rather than holomorphy. In fact, the class $S_{2, gen}(B_Δ)$ consisting of functions that satisfy the inequality for those matrices enjoys \begin{equation*} d_{\mathbb{D}}(f(z), f(w))\le d_Δ(z, w) \;\;(z,w\in B_Δ, f\in S_{2, gen}(B_Δ)). \end{equation*} Here, $d_Δ$ is a function defined by a matrix $Δ$ of abstract functions. Later, we focus on the case when $Δ$ is a matrix of holomorphic functions. We use the pseudo-distance $d_Δ$ to give a sufficient condition on a diagonalizable commuting tuple $T$ acting on $\mathbb{C}^2$ for $B_Δ$ to be a complete spectral domain for $T$. We apply this sufficient condition to generalizing von Neumann's inequalities studied by Drury and by Hartz-Richter-Shalit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some relations between Schwarz-Pick inequality and von Neumann's inequality Kojin, Kenta Functional Analysis We study a Schwarz-Pick type inequality for the Schur-Agler class $SA(B_δ)$. In our operator theoretical approach, von Neumann's inequality for a class of generic tuples of $2\times 2$ matrices plays an important role rather than holomorphy. In fact, the class $S_{2, gen}(B_Δ)$ consisting of functions that satisfy the inequality for those matrices enjoys \begin{equation*} d_{\mathbb{D}}(f(z), f(w))\le d_Δ(z, w) \;\;(z,w\in B_Δ, f\in S_{2, gen}(B_Δ)). \end{equation*} Here, $d_Δ$ is a function defined by a matrix $Δ$ of abstract functions. Later, we focus on the case when $Δ$ is a matrix of holomorphic functions. We use the pseudo-distance $d_Δ$ to give a sufficient condition on a diagonalizable commuting tuple $T$ acting on $\mathbb{C}^2$ for $B_Δ$ to be a complete spectral domain for $T$. We apply this sufficient condition to generalizing von Neumann's inequalities studied by Drury and by Hartz-Richter-Shalit. |
| title | Some relations between Schwarz-Pick inequality and von Neumann's inequality |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2306.08694 |