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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.

In this paper, we study spectrums of Schauder operators. We show that we always can choose a Schauder operator in a given orbit such that the Schauder spectrum of it is empty.