Cowen-Douglas operators on quaternionic Hilbert spaces
Xiaoqi Feng, Bingzhe Hou, Kui Ji
TL;DR
This work extends the Cowen-Douglas framework to quaternionic Hilbert spaces by defining the class $B_n^{s}(Ω_q)$ through the $S$-spectrum and showing each operator in this class yields an $n$-dimensional Hermitian right holomorphic quaternionic vector bundle over the axially reduced domain $Ω_{red}$. It establishes a rigidity principle: two operators in $B_n^{s}(Ω_q)$ are quaternion unitarily equivalent if and only if their associated bundles are equivalent, and it provides a canonical matrix description for $B_1^{s}(Ω_q)$ that yields a complete unitary classification via axial conjugation. The paper also demonstrates that curvature is not a complete unitary invariant in the quaternionic setting and proves that unitary equivalence of operators corresponds to unitary equivalence of their complex representations, with explicit examples including backward unilateral weighted shifts. Overall, it builds a quaternionic geometric-operator theory connecting $S$-spectral data to holomorphic quaternionic vector bundles and their unitary classifications.
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 $B_n^s(Ω_q)$. 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 $B_n^{s}(Ω_q)$ 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 $B_n^{s}(Ω_q)$ 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 $B_1^{s}(Ω_q)$ and furthermore, we give the quaternion unitarily equivalent classification of $B_1^{s}(Ω_q)$ 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 $B_1^{s}(Ω_q)$ are quaternion unitarily equivalent if and only if their complex representations are unitarily equivalent. Some relevant examples of the above results are also provided.