Cowen-Douglas operators on quaternionic Hilbert spaces

TL;DR

This paper extends the Cowen-Douglas class of operators to quaternionic Hilbert spaces using the $S$-spectrum, establishing classification results and exploring geometric properties unique to the quaternionic setting.

Contribution

It generalizes Cowen-Douglas operators to quaternionic Hilbert spaces, introduces canonical matrix representations, and analyzes their unitary equivalence and geometric invariants.

Findings

Operators in $B_1^{s}(\Omega_q)$ are classified by canonical matrix representations.

Quaternionic Cowen-Douglas operators are not fully characterized by curvature alone.

Two operators with the same curvature may not be unitarily equivalent in the quaternionic setting.

Abstract

In 1978, M. J. Cowen and R. G. Douglas introduced a class of geometric operators (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such operators. 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.