In 1978, Cowen and Douglas introduced a class of geometric operators (known as Cowen–Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such operators. They gave a complete set of unitary invariants in terms of the curves and its covariant derivatives. In this paper, after giving some basic properties of S-spectrum and right eigenvalues of bounded right linear operators on separable quaternionic Hilbert spaces, we generalize the class of Cowen–Douglas operators to the quaternionic Hilbert space via the S-spectrum and denote this class as \(B_n^s(\Omega _q)\) . Due to the lack of commutativity of quaternion multiplication, the quaternionic Cowen–Douglas operators are not trivial generalizations of the classical Cowen–Douglas operators. Each operator in \(B_n^{s}(\Omega _q)\) corresponds to an n-dimensional Hermitian right holomorphic quaternionic vector bundle. We first establish a rigidity theorem for Hermitian right holomorphic quaternionic vector bundles. It is then proven that two operators in \(B_n^{s}(\Omega _q)\) are quaternion unitarily equivalent if and only if the associate bundles are equivalent as Hermitian right holomorphic quaternionic vector bundles. In particular, we introduce canonical matrix representations of operators in \(B_1^{s}(\Omega _q)\) and furthermore, we give the quaternion unitarily equivalent classification of \(B_1^{s}(\Omega _q)\) by the canonical matrix representations. It is worth noting that curvature is a complete unitary invariant for the classical (complex) Cowen–Douglas operators, however, there exist two quaternionic Cowen–Douglas operators which have the same curvature but are not quaternion unitarily equivalent. In addition, we prove that the operators in \(B_1^{s}(\Omega _q)\) are quaternion unitarily equivalent if and only if their complex representations are unitarily equivalent. Some relevant examples of the above results are also provided.
