On von Neumann's inequality on the polydisc
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915008328237056 |
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| author | Hartz, Michael |
| author_facet | Hartz, Michael |
| contents | Given a $d$-tuple $T$ of commuting contractions on Hilbert space and a polynomial $p$ in $d$-variables, we seek upper bounds for the norm of the operator $p(T)$. Results of von Neumann and Andô show that if $d=1$ or $d=2$, the upper bound $\|p(T)\| \le \|p\|_\infty$, holds, where the supremum norm is taken over the polydisc $\mathbb{D}^d$. We show that for $d=3$, there exists a universal constant $C$ such that $\|p(T)\| \le C \|p\|_\infty$ for every homogeneous polynomial $p$. We also show that for general $d$ and arbitrary polynomials, the norm $\|p(T)\|$ is dominated by a certain Besov-type norm of $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14548 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On von Neumann's inequality on the polydisc Hartz, Michael Functional Analysis Complex Variables Primary 47A13, Secondary 47A30, 47A60 Given a $d$-tuple $T$ of commuting contractions on Hilbert space and a polynomial $p$ in $d$-variables, we seek upper bounds for the norm of the operator $p(T)$. Results of von Neumann and Andô show that if $d=1$ or $d=2$, the upper bound $\|p(T)\| \le \|p\|_\infty$, holds, where the supremum norm is taken over the polydisc $\mathbb{D}^d$. We show that for $d=3$, there exists a universal constant $C$ such that $\|p(T)\| \le C \|p\|_\infty$ for every homogeneous polynomial $p$. We also show that for general $d$ and arbitrary polynomials, the norm $\|p(T)\|$ is dominated by a certain Besov-type norm of $p$. |
| title | On von Neumann's inequality on the polydisc |
| topic | Functional Analysis Complex Variables Primary 47A13, Secondary 47A30, 47A60 |
| url | https://arxiv.org/abs/2311.14548 |