De Branges–Rovnyak spaces and local Dirichlet spaces of higher order

Abstract

Abstract We discuss de Branges–Rovnyak spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ ( b ) {Y=S|_{\mathcal{H}(b)}} , the restriction of the shift operator S on H 2 {H^{2}} to ℋ ⁢ ( b ) {\mathcal{H}(b)} , is a strict 2 ⁢ m {2m} -isometry and proved that such spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ ( b ) {\mathcal{H}(b)} coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213].