Left invertible quasi-isometric liftings

TL;DR

The paper investigates left invertible quasi-isometric liftings for operators similar to contractions, establishing a range-based, operator-inequality characterization via an invertible $A$ with $T^*T\le A$ and $T^*AT\le A$ together with $\mathcal{R}[(A-T^*T)^{1/2}]=\mathcal{R}[(A-T^*AT)^{1/2}]$. It shows that, when such $A$ exists, the lifting $S$ can be realized with a concrete block structure, and it details how the matrix form simplifies for quasicontractions. The work further proves that every operator similar to a contraction admits a left invertible quasi-isometric lifting that is similar to an isometry, though not all operators admit a left invertible natural lifting; a counterexample is given. Collectively, the results extend dilation-type techniques to a broader class of liftings, clarifying when liftings resemble isometric liftings and how their block matrices encode intertwinings with $A$-contractions.

Abstract

Quasi-isometric liftings similar to isometries, for the operators similar to contractions in Hilbert spaces, are investigated. The existence of such liftings is established, and their applications are explored for specific operator classes, including quasicontractions. A particular focus is placed on operators that admit left invertible minimal quasi-isometric liftings. These operators are characterized within the framework of $A$-contractions, and the matrix structures of their liftings are analyzed, highlighting parallels with the isometric liftings of contractions.

Theorems & Definitions (28)

• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6
• Theorem 2.7