Geometric similarity invariants of Cowen-Douglas operators

[摘要] In 1978, M. J. Cowen and R.G. Douglas introduced a class of operators Bn(Ω)B_n(\Omega)Bn​(Ω) (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in Bn(D)B_n(\mathbb{D})Bn​(D) can be constructed from a given operator in B1(D)B_1(\mathbb{D})B1​(D). We also characterize a class of strongly irreducible weakly homogeneous operators in Bn(D)B_n(\mathbb{D})Bn​(D).

[学科分类] 神经科学

[关键词] Curvature;second fundamental form;similarity;Property (H)