Journal of Operator Theory

Volume 61, Issue 1, Winter 2009 pp. 87-118.

Schur multipliers and de Branges--Rovnyak spaces: the multiscale case

Authors: Daniel Alpay (1), Aad Dijksma (2), and Dan Volok (3)
Author institution: (1) Department of Mathematics, Ben--Gurion University of the Negev, Beer-Sheva, 84105, Israel
(2) Department of Mathematics, The University of Groningen, Groningen, NL 9700AV, The Netherlands
(3) Department of Mathematics, Kansas State University, 138 Cardwell Hall, Manhattan, KS,66506, USA

Summary: We consider bounded linear operators acting on the $\ell_2$ space indexed by the nodes of a homogeneous tree. Using the Cuntz relations between the primitive shifts on the tree, we generalize the notion of the single-scale time-varying point evaluation and introduce the corresponding reproducing kernel Hilbert space in which Cauchy's formula holds. These notions are then used in the study of the Schur multipliers and of the associated de Branges-Rovnyak spaces. As an application we obtain realization of Schur multipliers as transfer operators of multiscale input-state-output systems.

Keywords: Non-commutative power series, system realization, homogeneous tree

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