A lifting theorem for 3-isometries

TL;DR

This paper establishes a new lifting theorem characterizing 3-isometries as restrictions of Jordan operators, connecting operator polynomial positivity with structural operator decompositions.

Contribution

It introduces a lifting theorem for 3-isometries, linking positivity conditions to their realization as restrictions of Jordan operators, extending previous results for 3-symmetric operators.

Findings

Characterization of 3-isometries as restrictions of Jordan operators

Positivity conditions equivalent to operator decompositions

Extension of known results for 3-symmetric operators

Abstract

An operator T on Hilbert space is a 3-isometry if there exists operators B and D such that (T*)^n T^n = I+nB +n^2 D. An operator J is a Jordan operator if it the sum of a unitary U and nilpotent N of order two which commute. If T is a 3-isometry and c>0, then I-c^{-2} D + sB + s^2D is positive semidefinite for all real s if and only if T is the restriction to an invariant subspace of a Jordan operator J=U+N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.