Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator $H_{\theta,\lambda} =U+U^*+(\lambda /2)(V+V^*)$ in the rotation $C^*$-algebra $A_\theta =C^*(U,V unitaries;UV=e^{2\pi i\theta} VU)$. In this process, we significantly improve the inequality $\| H_\theta \| \leq 2\sqrt{2}$, $\theta \in [0.25,0.5]$, conjectured by Beguin, Valette and Zuk.