Cowen-Douglas operators on quaternionic Hilbert spaces
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arXiv
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2025
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| _version_ | 1866914107944337408 |
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| author | Feng, Xiaoqi Hou, Bingzhe Ji, Kui |
| author_facet | Feng, Xiaoqi Hou, Bingzhe Ji, Kui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cowen-Douglas operators on quaternionic Hilbert spaces Feng, Xiaoqi Hou, Bingzhe Ji, Kui Functional Analysis Operator Algebras 47B13, 47S05, 32L05 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. |
| title | Cowen-Douglas operators on quaternionic Hilbert spaces |
| topic | Functional Analysis Operator Algebras 47B13, 47S05, 32L05 |
| url | https://arxiv.org/abs/2510.19523 |