A dilation theoretic approach to Banach spaces

TL;DR

This paper characterizes when a Banach space is a Hilbert space using dilation properties of contractions, providing new criteria, explicit dilations, and examples distinguishing Hilbert from non-Hilbert spaces.

Contribution

It introduces novel dilation-based criteria for identifying Hilbert spaces among Banach spaces and characterizes contractions that dilate to isometries, with explicit constructions.

Findings

Banach space is Hilbert iff all strict contractions dilate to isometries

Constructed examples of non-Hilbert spaces with non-dilating contractions

Characterized contractions that dilate to isometries on non-Hilbert spaces

Abstract

For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the function $A_T: \mathbb X \rightarrow [0, \infty]$ defined by $A_T(x)=(\|x\|^2 -\|Tx\|^2)^{\frac{1}{2}}$ gives a norm on $\mathbb X$. We also find several other necessary and sufficient conditions in this thread such that a Banach sapce becomes a Hilbert space. We construct examples of strict contractions on non-Hilbert Banach spaces that do not dilate to isometries. Then we characterize all strict contractions on a non-Hilbert Banach space that dilate to isometries and find explicit isometric dilation for them. We prove several other results including characterizations of complemented subspaces in a Banach space, extension of a Wold isometry to a Banach space unitary and describing norm attainment sets of Banach space operators in terms of dilations.