Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 93-120.

On the metric of the jet bundle and similarity of Cowen--Douglas operators

Authors: Kui Ji (1), Shanshan Ji (2), Hyun-Kyoung Kwon (3), Jing Xu (4)
Author institution: (1) School of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050024, China
(2) College of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang, Hebei 050061, China
(3) Department of Mathematics and Statistics, University at Albany-State University of New York, Albany, NY, 12222, U.S.A.
(4) School of Mathematics and Science, Hebei GEO University, Shijiazhuang 052161, China

Summary: The study of Cowen-Douglas operators involves traditional operator-theoretic and complex-geometric tools. We study the similarity of Cowen-Douglas operators to backward shifts on analytic function spaces whose multiplier algebras equal $H^\infty(\mathbb D)$. Using the ratio of metrics and a model theorem, we obtain a sufficient condition for similarity. For weighted Dirichlet spaces, whose multiplier algebra is a proper subalgebra of $H^\infty(\mathbb D)$, we give a similarity condition via the jet bundle. The analysis relies on M\"uller’s model theory for Dirichlet shifts and the corona theorem of Kidane-Trent for weighted Dirichlet spaces.

DOI: http://dx.doi.org/10.7900/jot.2024dec18.2482
Keywords: Dirichlet space, model theorem, multipliers, weights, backward shift operator, similarity

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