Regular quantum annulus unitary dilation and applications

Abstract

Consider the annulus $\mathbb{A}_r=\{z\in\mathbb{C}:r^{-1}<|z|<r\}$ for $r>1$ and the quantum annulus \[ Q\mathbb{A}_r=\{T:\ T \text{ is an invertible operator and} \ \|T\|, \|T^{-1}\|\leq r\}. \] McCullough and Pascoe proved that $T\in Q\mathbb{A}_r$ if and only if $β(T^*,T)=(r^2+r^{-2})-T^*T-(T^*T)^{-1}\ge0$. We call an invertible operator $T$ a quantum annulus unitary if $β(T^*,T)=0$. In this article, we construct an explicit doubly commuting $d$-tuple of quantum annulus unitaries that simultaneously extends a given doubly commuting $d$-tuple of operators in $Q\mathbb{A}_r$. We introduce the notion of a regular quantum annulus unitary dilation and show that the dilation arising from our construction is regular. As an application of the dilation theorem, we show that $\overline{\mathbb A}_r$ is a complete $K_t$-spectral set for operators in $Q\mathbb{A}_r$ and $\overline{\mathbb{A}}_r^d$ is a complete $K_{dc}^{(d)}$-spectral set for doubly commuting $d$-tuples of operators in $Q\mathbb{A}_r$, where \[ K_t=2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right) \quad \text{and} \quad K_{dc}^{(d)}=\left[2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right)\right]^d. \] We further prove that every doubly commuting tuple of operators in $Q\mathbb{A}_r$ is similar to a commuting tuple having $\overline{\mathbb{A}}_r^{d}$ as a complete spectral set. In addition, we establish bounds for the optimal spectral constants and show that they converge to $2^d$ as $r\to\infty$. We also obtain an alternative characterization of operators in $Q\mathbb{A}_r$ and quantum annulus unitaries, and prove that $\overline{\mathbb{A}}_r^d$ is a $K$-spectral set for a subclass of commuting $d$-tuples in $Q\mathbb{A}_r$.