A dilation theoretic approach to Banach 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 $A_T:\mathbb{X}\to [0,\infty ]$ defined by $A_T(x)=(\Vert x{\Vert }^2-\Vert Tx{\Vert }^2{)}^{\frac{1}{2}}$ 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.

Revised, A new section added, Submitted to journal