GEOMETRY OF HOLOMORPHIC VECTOR BUNDLES AND SIMILARITY OF COMMUTING TUPLES OF OPERATORS

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(2009), 135-165 about the similarity of quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of Cowen-Douglas class is obtained.