A transference principle for involution-invariant functional Hilbert spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908425610330112 |
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| author | Bera, Santu Chavan, Sameer Jain, Shubham |
| author_facet | Bera, Santu Chavan, Sameer Jain, Shubham |
| contents | Let $σ: \mathbb C^d \rightarrow \mathbb C^d$ be an affine-linear involution such that $J_σ= -1$ and let $U, V$ be two domains in $\mathbb C^d.$ Let $ϕ: U \rightarrow V$ be a $σ$-invariant $2$-proper map such that $J_ϕ$ is affine-linear and let $\mathscr H(U)$ be a $σ$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space $\mathscr H_ϕ(V):=\{f \in \mathrm{Hol}(V) : J_ϕ\cdot f \circ ϕ\in \mathscr H(U)\}$ endowed with the norm $\|f\|_ϕ:=\|J_ϕ\cdot f \circ ϕ\|_{\mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear mapping $\varGamma_ϕ$ defined by $ \varGamma_ϕ(f) = J_ϕ\cdot f \circ ϕ,$ $f \in \mathrm{Hol}(V),$
is a unitary from $\mathscr H_ϕ(V)$ onto $\{f \in \mathscr H(U) : f = -f \circ σ\}.$ Moreover, a neat formula for the reproducing kernel $κ_ϕ$ of $\mathscr H_ϕ(V)$ in terms of the reproducing kernel of $\mathscr H(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann's inequality for contractive tuples naturally associated with these domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04384 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A transference principle for involution-invariant functional Hilbert spaces Bera, Santu Chavan, Sameer Jain, Shubham Complex Variables Functional Analysis Primary 32A36, 47A13, Secondary 32A10, 46E22 Let $σ: \mathbb C^d \rightarrow \mathbb C^d$ be an affine-linear involution such that $J_σ= -1$ and let $U, V$ be two domains in $\mathbb C^d.$ Let $ϕ: U \rightarrow V$ be a $σ$-invariant $2$-proper map such that $J_ϕ$ is affine-linear and let $\mathscr H(U)$ be a $σ$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space $\mathscr H_ϕ(V):=\{f \in \mathrm{Hol}(V) : J_ϕ\cdot f \circ ϕ\in \mathscr H(U)\}$ endowed with the norm $\|f\|_ϕ:=\|J_ϕ\cdot f \circ ϕ\|_{\mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear mapping $\varGamma_ϕ$ defined by $ \varGamma_ϕ(f) = J_ϕ\cdot f \circ ϕ,$ $f \in \mathrm{Hol}(V),$ is a unitary from $\mathscr H_ϕ(V)$ onto $\{f \in \mathscr H(U) : f = -f \circ σ\}.$ Moreover, a neat formula for the reproducing kernel $κ_ϕ$ of $\mathscr H_ϕ(V)$ in terms of the reproducing kernel of $\mathscr H(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. Although some of these are known, this allows us to obtain an analog of von Neumann's inequality for contractive tuples naturally associated with these domains. |
| title | A transference principle for involution-invariant functional Hilbert spaces |
| topic | Complex Variables Functional Analysis Primary 32A36, 47A13, Secondary 32A10, 46E22 |
| url | https://arxiv.org/abs/2408.04384 |