A von Neumann type inequality for an annulus

Abstract: Let $A_r={r<|z|<1}$ be an annulus. We consider the class of operators $\mathcal{F}r:={T\in\mathcal{B}(H): r^2T^{-1}(T^{-1})^+TT^\le r^2+1,\hspace{0.08 cm}\sigma(T)\subset A_r}$ and show that for every bounded holomorphic function $\phi$ on $A_r:$ $$\sup{T\in\mathcal{F}r}||\phi(T)||\le\sqrt{2}||\phi||{\infty},$$ where the constant $\sqrt{2}$ is the best possible. We do this by characterizing the calcular norm induced on $H^{\infty}(A_r)$ by $\mathcal{F}_r$ as the multiplier norm of a suitable holomorphic function space on $A_r$.