A von Neumann type inequality for an annulus

Abstract

Let $A_r=\{r<|z|<1\}$ be an annulus. We consider the class of operators ${\mathcal{F}}_r:=\{T\in \mathcal{B}(H):r^2{T}^{-1}(T^{-1}{)}^{\ast }+T{T}^{\ast }\le r^2+1,\sigma (T)\subset A_r\}$ and show that for every bounded holomorphic function $\varphi$ on $A_r:$ $$\underset{T\in {\mathcal{F}}_r}{\sup }||\varphi (T)||\le \sqrt{2}||\varphi |{|}_{\infty },$$ 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 $A_r$.

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