On a minimal And\^{o} dilation for a pair of strict contractions

TL;DR

This paper develops a minimal isometric dilation for pairs of strict contractions, improving upon Andô's dilation, and extends the concept to Banach and normed spaces under certain conditions.

Contribution

It constructs a minimal Andô dilation for commuting strict contractions on Hilbert spaces and extends the approach to Banach and normed spaces with specific properties.

Findings

Constructed a minimal isometric dilation for strict contractions on Hilbert spaces.

Extended dilation construction to Banach space contractions under certain conditions.

Showed dilation possibility for more general pairs of contractions on normed spaces.

Abstract

The isometric dilation of a pair of commuting contractions due to And\^{o} is not minimal. We modify And\^{o}'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 And\^{o} dilation for a commuting pair of strict Banach space contractions. Further, we show that an And\^{o} 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 $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.