Harnack and Shmul'yan pre-order relations for Hilbert space contractions

TL;DR

This paper investigates the relationships between Harnack and Shmul'yan pre-orders for Hilbert space contractions, establishing conditions for their equivalence and exploring applications to special classes of operators.

Contribution

It provides new conditions linking Harnack and Shmul'yan equivalences and extends the Shmul'yan pre-order to operator-valued Schur class functions.

Findings

Harnack and Shmul'yan equivalences coincide under certain conditions.

The parts of partial isometries with respect to Harnack and Shmul'yan are identical.

Explicit description of the Shmul'yan-ter Horst part of a partial isometry.

Abstract

We study the behavior of some classes of Hilbert space contractions with respect to Harnack and Shmul'yan pre-orders and the corresponding equivalence relations. We give some conditions under which the Harnack equivalence of two given contractions is equivalent to their Shmul'yan equivalence and to the existence of an arc joining the two contractions in the class of operator-valued contractive analytic functions on the unit disc. We apply some of these results to quasi-isometries and quasi-normal contractions, as well as to partial isometries for which we show that their Harnack and Shmul'yan parts coincide. We also discuss an extension, recently considered by S.~ter~Horst [\emph{J. Operator Th. 72(2014), 487--520}], of the Shmul'yan pre-order from contractions to the operator-valued Schur class of functions. In particular, the Shmul'yan-ter Horst part of a given partial isometry, viewed as a constant Schur class function, is explicitly determined.