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 ${\hat{I}}^2(T^{\ast },T)=(r^2+r^{-2})-T^{\ast }T-(T^{\ast }T{)}^{-1}\ge 0$. We call an invertible operator $T$ a quantum annulus unitary if ${\hat{I}}^2(T^{\ast },T)=0$. 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 $Q{\mathbb{A}}_r$.

28 Pages