On a minimal Andô dilation for a pair of strict contractions

Abstract

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal{H}{\oplus }_2{\ell }_2(\mathcal{H}{\oplus }_2\mathcal{H})$ for a commuting pair of strict contractions on a Hilbert space $\mathcal{H}$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb{X}$ provided that the function $A_{T_i}:\mathbb{X}\to \mathbb{R}$ given by $A_{T_i}(x)=(\Vert x{\Vert }^2-\Vert T_{i}x{\Vert }^2{)}^{\frac{1}{2}}$ defines a norm on $\mathbb{X}$ for $i=1,2$.

15 pp, Proceedings of the American Mathematical Society, To appear