Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 169-202.

Geometry of holomorphic vector bundles and similarity of commuting tuples of operators

Authors: Yingli Hou (1), Kui Ji (2), Shanshan Ji (3) Jing Xu (4)
Author institution: (1) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China
(2) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China
(3) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China
(4) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China

Summary: In this paper, a new criterion for the similarity of commuting tuples of operators on Hilbert spaces is introduced. As an application, we obtain a geometric similarity invariant of tuples in the Cowen-Douglas class which gives a partial answer to a question raised by R.G. Douglas in Complex geometry and operator theory, Acta Math. 141(1978), 187-261 and Operator theory and complex geometry, Extracta Math. 24(2007), 135-165 about the similarity of quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of Cowen-Douglas class is obtained.

DOI: http://dx.doi.org/10.7900/jot.2022mar04.2378
Keywords: commuting tuple, Cowen-Douglas operator, curvature, holomorphic bundle, similarity

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