Dilation of normal operators associated with an annulus

Abstract

For $0<r<1$, let us consider the following annulus: \[ \mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}. \] A Hilbert space operator $T$ for which ${\overline{\mathbb{A}}}_r$ is a spectral set is called an ${\mathbb{A}}_r$-\textit{contraction}. Also, a normal operator $U$ whose spectrum lies on the boundary $\partial {\mathbb{A}}_r$ of ${\mathbb{A}}_r$ is called an ${\mathbb{A}}_r$-\textit{unitary}. We prove that any $m$ number of commuting normal ${\mathbb{A}}_r$-contractions $N_1,\dots ,N_m$ can be simultaneously dilated to commuting ${\mathbb{A}}_r$-unitaries $U_1,\dots ,U_m$. To construct such a dilation, we solve a Dirichlet problem for the polyannulus ${\mathbb{A}}_r^m$. Also, we show that any finitely many doubly commuting subnormal ${\mathbb{A}}_r$-contractions simultaneously dilate to commuting ${\mathbb{A}}_r$-unitaries. Finally, we show that such a simultaneous ${\mathbb{A}}_r$-unitary dilation holds for any finite number of doubly commuting $2\times 2$ scalar ${\mathbb{A}}_r$-contractions.

arXiv comment22 Pages