A dilation theoretic approach to Banach spaces

TL;DR

The paper develops a comprehensive dilation theory for contractions on complex Banach spaces and shows that all strict contractions dilate to isometries if and only if the space is a Hilbert space, tying this to the norm defined by $A_T(x) = (\|x\|^2 - \|Tx\|^2)^{1/2}$. It constructs explicit minimal isometric dilations, generalizes Schäffer-type dilations to Banach spaces, and identifies the minimal dilation space as $\mathbb{X} \oplus_2 \ell_2(\mathbb{X}_0)$ with $\mathbb{X}_0=(\mathbb{X},A_T)$. The work introduces a Banach-space adjoint $T_*$, develops a rich theory of complemented subspaces, Wold-type decompositions, and $\sigma$-shifts, and connects dilation to norm attainment sets and spectral properties. Together, these results yield new characterizations of Hilbert spaces, extend canonical Hilbert-space dilation theory to Banach spaces, and provide a unified framework linking dilation, duality, and subspace structure in Banach-space operator theory.

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.

Theorems & Definitions (148)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• Definition 2.3: Fackler & Glük, SJ
• Theorem 2.4