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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