arXiv subjects

Kui Ji

Publications and source records attributed to Kui Ji.

At least 19 recordsLinked to original sources

arXiv2025

Cowen-Douglas operators on quaternionic Hilbert spaces

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.

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arXiv2025

Cyclicity of Cowen-Douglas tuples

The study of Cowen-Douglas operators involves not only operator-theoretic tools but also complex geometry on holomorphic vector bundles. By leveraging the properties of holomorphic vector bundles, this paper investigates the cyclicity of Cowen-Douglas tuples and demonstrates conclusively that every such tuple is cyclic.

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arXiv2024

On the metric of the jet bundle and similarity of Cowen-Douglas operators

The study of Cowen-Douglas operators not only involves traditional operator-theoretic tools but also concepts and results from complex geometry on holomorphic vector bundles. We make use of the ratio of the metric matrices first considered by Clark and Misra and a model theorem by Agler to describe the similarity of backward shift operators on analytic function spaces whose multiplier algebras are the space of bounded analytic functions. It is well-known that, in general, it becomes much more complicated to formulate a sufficient condition for similarity than a necessary one. We also give a sufficient condition for a Cowen-Douglas operator to be similar to the backward shift operator on the Dirichlet space with weights by introducing a condition on the jet bundle of a holomorphic vector bundle. Note that the multiplier algebras of these spaces do not coincide with the space of bounded, analytic functions as in other analytic functions spaces, requiring a different approach. The results by M\"{u}ller on operator models related to Dirichlet shifts and by Kidane and Trent on the corona problem for the multiplier algebras of weighted Dirichlet spaces are indispensable tools in attaining this similarity result.

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arXiv2023

On the similarity of powers of operators with flag structure

Let $\mathrm{L}^2_a(\mathbb{D})$ be the classical Bergman space and denote $M_h$ for the multiplication operator by a function $h$. Let $B$ be a finite Blaschke product with order $n$.An open question proposed by R. G. Douglas is whether the operators $M_B$ on $\mathrm{L}^2_a(\mathbb{D})$ similar to $\oplus_1^n M_z$ on $\oplus_1^n \mathrm{L}^2_a(\mathbb{D})$? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernels.Since the operator $M_z^*$ is in Cowen-Douglas class $B_1(\mathbb{D})$ in many cases, Douglas's question can be expressed as a version for operators in $B_1(\mathbb{D})$, and it is affirmative for many operators in $B_1(\mathbb{D})$.A natural question is how about Douglas's question in the version for operators in Cowen-Douglas class $B_n(\mathbb{D})$ ($n>1$)? In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class $B_2(\mathbb{D})$, and give a negative answer.This indicates that Douglas's question cannot be directly generalized to general Hilbert spaces with vector-valued analytical reproducing kernel.

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arXiv2023

The Cowen-Douglas operators with strongly flag structure

Denote $\mathcal{FB}_{n}(\Omega)$ as the collection of operators possessing a flag structure in the Cowen-Douglas class $\mathcal{B}_{n}(\Omega)$, and all the irreducible homogeneous operators in $\mathcal{B}_{n}(\Omega)$ belong to this class. G. Misra et al. pointed out in \cite{JJKM} that the unitary invariants of this class of operators include the curvature and the second fundamental form of the corresponding line bundle. In terms of the invariants, it is more tractable compared to general operators in $\mathcal{B}_{n}(\Omega)$. A subclass of $\mathcal{FB}_{n}(\Omega)$, denoted by $\mathcal{CFB}_{n}(\Omega)$, was proven to be norm dense in $\mathcal{B}_{n}(\Omega)$ in \cite{JJ}. In this paper, we introduce a smaller subclass of $\mathcal{FB}_{n}(\Omega)$ which possesses a strongly flag structure, and for which the curvature and the second fundamental form of the associated line bundle is a complete set of unitary invariants. And we notice that this class of operators is norm dense in $\mathcal{B}_{n}(\Omega)$ up to similarity. On this basis, we have completed the similar classification of a large class of operators with flag structure, which reduces the number of the similarity invariants in \cite{JKSX} from $\frac{n(n-1)}{2}+1$ to $n$. Furthermore, we also get a complete characterization of weakly homogeneous operators with high index and flag structure.

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arXiv2022

The Cowen-Douglas Theory for Operator Tuples and Similarity

We are concerned with the similarity problem for Cowen-Douglas operator tuples. The unitary equivalence counterpart was already investigated in the 1970's and geometric concepts including vector bundles and curvature appeared in the description. As the Cowen-Douglas conjecture show, the study of the similarity problem has not been so successful until quite recently. The latest results reveal the close correlation between complex geometry, the corona problem, and the similarity problem for single Cowen-Douglas operators. Without making use of the corona theorems that no longer hold in the multi-variable setting, we prove that the single operator results for similarity remain true for commuting Cowen-Douglas operator tuples as well.

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arXiv2021

A note on unitary equivalence of operators acting on reproducing kernel Hilbert spaces

A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.

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arXiv2020

On the similarity of restriction of the operator to an invariant subspace

Let $M_{z}$ be the multiplication operator on the Bergman space and $M_{I}$ denote the restriction of $M_{z}$ to an invariant subspace $I$. A question raised by K. Zhu is that when are two restriction operators $M_{I}$ and $M_{J}$ are similar? In this note, we give some sufficient conditions of this problem in a general case.

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arXiv2020

Geometric Similarity invariants of Cowen-Douglas Operators

In 1978, M. J. Cowen and R.G. Douglas introduce a class of operators (known as Cowen-Douglas class of operators) and associates a Hermitian holomorphic vector bundle to such an operator in a very influential paper. They give a complete set of unitary invariants in terms of involving the curvature and its covariant partial derivatives. At the same time they ask: can one use geometric ideas to characterize completely the similarity invariants of Cowen-Douglas operators? We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely characterize the similarity invariants for a norm dense class of Cowen-Douglas operators.

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arXiv2019

A subclass of the Cowen-Douglas class and similarity

We consider a subclass of the Cowen-Douglas class in which the problem of deciding whether two operators are similar becomes more manageable. A similarity criterion for Cowen-Douglas operators is known to be dependent on the trace of the curvatures of the corresponding eigenvector bundles. Unless the given eignvector bundle is a line bundle, the computation of the curvatures, in general, is not so simple as one might hope. By using a structure theorem given in \cite{JW}, we reduce the problem of finding the trace of the curvatures to looking at the curvatures of the associated line bundles. Moreover, several questions related to the similarity problem are also taken into account.

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arXiv2019

Similarity Invariants of Essentially normal Cowen-Douglas Operators and Chern Polynomials

In this paper, we systematically study a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and prove a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

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arXiv2019

Cowen-Douglas operators and the third of Halmos' ten problems

Let $T$ be a bounded linear operator on a complex separable infinite dimensional Hilbert space $\mathcal{H}$. $T$ is called intransitive if it leaves invariant spaces other than 0 or the whole space $\mathcal{H}$; otherwise it is transitive. In 1970, P. R. Halmos raised ten open problems on operator theory. In the past more than 50 years, nine of Halmos' ten problems were answered, but only the third one has made little progress. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? In this paper, we establish a set of theoretical systems with the help of Cowen-Douglas operators and spectral analysis. We give an affirmative answer to this problem under certain spectral conditions, which make essential progress in the research of Halmos' third problem. As the first application, we show that for an invertible hyponormal operator $T$, if $T^{-1}$ is intransitive and int$\sigma(T^{-1})^{\land}$ is not connected, then $T$ is also intransitive. As the second application, we show that if $T^{-1}$ has a proper strictly cyclic invariant subspace and there exists a bounded open set $\Omega$ which is a connected component of $\rho(T^{-1})$ such that $\Omega\cap \mathcal{U}_0=\emptyset$, where $\mathcal{U}_0$ is the connected component of $int(\sigma(T^{-1})^\land)$ containing zero point, then $T$ is intransitive.

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arXiv2019

$N$-hypercontractivity and similarity of Cowen-Douglas operators

When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the $n$-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.

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arXiv2018

Curvature formulas of holomorphic curves on $C^*$-algebras and Cowen-Douglas Operators

For $Ω\subseteq \mathbb{C}$ a connected open set, and ${\mathcal U}$ a unital $C^*$-algebra, let ${\mathcal I} ({\mathcal U})$ and ${\mathcal P}({\mathcal U})$ denote the sets of all idempotents and projections in ${\mathcal U}$ respectively. ${\mathcal P}({\mathcal U})$ is called as the Grassmann manifold of $\mathcal U$ and ${\mathcal I} ({\mathcal U})$ is called as the extended Grassmann manifold. If $P:Ω\rightarrow {\mathcal P}({\mathcal U})$ is a real-analytic ${\mathcal U}$-valued map which satisfies $\overline{\partial} PP=0$, then $P$ is called a holomorphic curve on ${\mathcal P}({\mathcal U})$. In this note, we will define the formulaes of curvature and it's covariant derivatives for holomorphic curves on $C^*$-algebras. It can be regarded as the generalization of curvature and it's covariant derivatives of the classical holomorphic curves. By using the curvature formulae, we give the unitarily and similarity classifications for the holomorphic curves and extended holomorphic curves on $C^*$-algebras respectively. And we also give a description of the trace of the covariant derivatives of curvature for any Hermitian holomorphic vector bundles. As applications, we also discuss the relationship between holomorphic curves, extended holomorphic curves, similarity of holomorphic Hermitian vector bundles and similarity of Cowen-Douglas operators.

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arXiv2018

Geometry of holomorphic vector bundles and similarity of commuting operator tuples

In this paper, a new criterion for the similarity of commuting tuples of operators on Hilbert spaces is introduced. As an application, we obtain a geometric similarity invariant of tuples in the Cowen-Douglas class which gives a partial answer to a question raised by R.G. Douglas about the similarity of quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of Cowen-Douglas class is obtained.

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arXiv2017

Similarity of Quotient Hilbert modules in the Cowen-Douglas Class

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and sufficient conditions for a class of Cowen-Douglas Hilbert modules satisfying some positivity conditions to be similar to $\mathcal{M} \otimes \mathbb{C}^m$. We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen-Douglas Hilbert module is quasi-affinity to a submodule of the free module $\mathcal{M} \otimes \mathbb{C}^m$.

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arXiv2014

Curvature and the Second fundamental form in classifying quasi-homogeneous holomorphic curves and operators in the Cowen-Douglas class

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen-Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen-Douglas class obtained recently by the last author and A. Korányi. Specifically, the complex geometric invariants used for our classification are the curvature and the second fundamental forms inherent in the definition of a quasi-homogeneous operator. We show that these operators are irreducible and determine when they are strongly irreducible. Applications include the equality of the topological and algebraic K-group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos on similarity for these operators.

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arXiv2014

Rigidity of the flag structure for a class of Cowen-Douglas operators

The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid, that is, the unitary equivalence class of the operator and the flag structure determine each other. A complete set of unitary invariants, which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class, are obtained.

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