Partial orders on partial isometries (Q2835242)

The authors study three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. It is shown that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries, these spaces can be realized as the well-known model subspaces and de Branges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate. The idea of transition from isometries to analytic functions goes back to \textit{M. S. Livshits} [Am. Math. Soc., Transl., II. Ser. 13, 85--103 (1960; Zbl 0089.10802)] and \textit{M. G. Krein} [Am. Math. Soc., Translat., II. Ser. 97, 75--143 (1971; Zbl 0258.47025)].