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

Swapan Jana; Sourav Pal

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arXiv · 2603.26321

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

Swapan Jana·Sourav Pal

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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 $i=1,2$.

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BibTeX RIS

Swapan Jana, Sourav Pal. 2026-03-27. On a minimal And\^{o} dilation for a pair of strict contractions. https://arxiv.org/abs/2603.26321

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