In this paper, on $\mathbb{D}$ we define Cowen-Douglas function introduced by Cowen-Douglas operator $M_ϕ^*$ on Hardy space $\mathcal{H}^2(\mathbb{D})$ and we give a sufficient condition for Cowen-Douglas function, where $ϕ\in\mathcal{H}^\infty(\mathbb{D})$. Moreover, we give some applications of Cowen-Douglas function on chaos, such as application on the inverse chaos problem for $ϕ(T)$, where $ϕ$ is a Cowen-Douglas function and $T$ is the backward shift operator on $\mathcal{L}^2(\mathbb{N})$.
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…
Geometric Similarity invariants of Cowen-Douglas Operators
1978年，M. J. Cowen和R.G. Douglas在一篇很有影响力的论文中引入了一类算子（称为Cowen-Douglas算子类），并将一个Hermitian全纯向量丛与这样的算子联系起来。他们给出了一套完整的酉不变量，涉及曲率及其协变偏导数。同时，他们问：可以使用几何的想法，完全特征的相似不变量的Cowen-Douglas运营商？我们对这个问题作了部分回答。本文证明了曲率和第二基本型完全刻画了范数稠密Cowen-Douglas算子类的相似不变量.
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. Mo…
It has been recently shown that if $K$ is a sesqui-analytic scalar valued non-negative definite kernel on a domain $Ω$ in $\mathbb C^m$, then the function $\big(K^2\partial_i\bar{\partial}_j\log K\big )_{i,j=1}^ m,$ is also a non-negative definite kernel on $Ω$. In this paper, we discuss two consequences of this result. The first one strengthens the curvature inequality for operators in the Cowen-Douglas class $B_1(Ω)$ while the second one gives a relationship of the reproducing kernel of a submodule of certain Hilbert modules with the curvature of the associated quotient module.
We construct a large family of positive-definite kernels $K: \mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\mbox{Möb} \times\cdots\times \mbox{Möb}$ ($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $\mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreduc…
In this article, we obtain two sets of results. The first set of complete results are exclusively for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc: * A classification of irreducible hermitian holomorphic vector bundles over $\mathbb{D}^2$, homogeneous with respect to $\mbox{Möb}\times \mbox{Möb}$, is obtained assuming that the associated representations are \textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\mathbb{D}^2$ of rank $1,2$ or $3$ is …
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 theor…
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.
We introduce a set of operator-valued invariants for localization of operator tuples in the Cowen-Douglas class, with which a Specht-type criterion for unitary equivalence is obtained.
Let $ Ω\subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(Ω)$ of $Ω$ dependin…
