A note on a spectral constant associated with an annulus

Fix $R>1$ and let $A_R=\{1/R\le |z|\le R \}$ be an annulus. Also, let $K(R)$ denote the smallest constant such that $A_R$ is a $K(R)$-spectral set for the bounded linear operator $T\in \mathcal{B}(H)$ whenever $||T||\le R$ and $||T^{-1}||\le R.$ We show that $K(R)\ge 2, \text{ for all } R>1.$ This improves on previous results by Badea, Beckermann and Crouzeix.