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Journal of Operator Theory

Volume 75, Issue 2, Spring 2016  pp. 409-442.

Partial orders on partial isometries

Authors:  Stephan Ramon Garcia (1), Robert T.W. Martin (2), and William T. Ross (3)
Author institution:(1) Department of Mathematics, Pomona College, Claremont, California, 91711 U.S.A.
(2) Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa
(3) Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, U.S.A.

Summary: This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show 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.

Keywords: Hardy space, model subspaces, de Branges-Rovnyak spaces, partial isometries, symmetric operators, partial order, pre-order

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