Geometry and operator theory on quaternionic Hilbert spaces

Abstract

In this article, we study the geometry and operator theory on quaternionic Hilbert spaces. As it is well-known, Cowen--Douglas operators are a class of non-normal operators related to complex geometry on complex Hilbert spaces. Our purpose is to generalize this concept on quaternionic Hilbert spaces. At the beginning, we study a class of complex holomorphic curves which naturally induce complex vector bundles as sub-bundles in the product space of the base space and a quaternionic Hilbert space. Then we introduce quaternionic Cowen--Douglas operators and give their quaternion unitarily equivalent invariant related to the geometry of the holomorphic curves.