A complete list of homogeneous operators in the Cowen-Douglas class $B_n(D)$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.
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 $\mathcal{L}^2(\mathbb{N})$.
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $\mathbb C^2$ this condition is always satisfied. As an application we show that all homogeneous pa…
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous $n$-tuples of op…
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.
In this paper we show a Cowen-Douglas operator $T \in \mathcal{B}_{n}(Ω)$ is the adjoint operator of some backward shift on a general basis by choosing nice cross-sections of its complex bundle $E_{T}$. Using the basis theory model, we show that a Cowen-Douglas operator never be a shift on some Markushevicz basis for $n \ge 2$.
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…
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…
Let T be a Cowen-Douglas tuple on a Banach space X. We use functional representations of T to associate with each T-invariant subspace Y of X an integer called the fiber dimension of Y. Among other results we prove a limit formula for the fiber dimension, show that it is invariant under suitable changes of Y and deduce a dimension formula for pairs of homogeneous invariant subspaces of graded Cowen-Douglas tuples on Hilbert spaces. We thus extend results proved by L. Chen, G. Cheng and X. Fang for single Cowen-Douglas operators on Hilbert spaces to the case of commuting operator systems on Ban…
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 qua…
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…
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.
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 …
In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation o…
Fix a bounded planar domain $Ω.$ If an operator $T,$ in the Cowen-Douglas class $B_1(Ω),$ admits the compact set $\barΩ$ as a spectral set, then the curvature inequality $\mathcal K_T(w) \leq - 4 π^2 S_Ω(w,w)^2,$ where $S_Ω$ is the Szego kernel of the domain $Ω,$ is evident. Except when $Ω$ is simply connected, the existence of an operator for which $\mathcal K_T(w) = 4 π^2 S_Ω(w,w)^2$ for all $w$ in $Ω$ is not known. However, one knows that if $w$ is a fixed but arbitrary point in $Ω,$ then there exists a bundle shift of rank $1,$ say $S,$ depending on this $w,$ such that $\mathcal K_{S^*}(w)…
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.
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}({\ma…
Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although the study of self-adjoint operators goes back a few decades, the non self-adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like $\mathcal A(Ω)$, where…
For any bounded domain $Ω$ in $\mathbb C^m,$ let ${\mathrm B}_1(Ω)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $\boldsymbol T$ in the Cowen-Douglas class ${\mathrm B}_1(Ω),$ let $N_{\boldsymbol T}(w)$ denote the restriction of $\boldsymbol T$ to the subspace ${\cap_{i,j=1}^m\ker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{\boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $ρ_{_{\!N_{\boldsymbol T}(w)}}(p) = p\big (N_{\boldsym…
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.
The main features of homogeneous Cowen-Douglas operators, well-known for the unit disk, are generalized to the setting of hermitian bounded symmetric domains of arbitrary rank.
A curvature inequality is established for contractive commuting tuples of operators in the Cowen-Douglas class of rank n. Properties of the extremal operators, that is, the operators which achieve equality, are investigated. Specifically, a substantial part of a well known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.
Let $Ω$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $Ω$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T=(T_1,\ldots, T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\mathbb{K}$-homogeneous $d$-tuple $\boldsym…
A bounded linear operator $T$ on a Hilbert space is said to be homogeneous if $φ(T)$ is unitarily equivalent to $T$ for all $φ$ in the group Möb of bi-holomorphic automorphisms of the unit disc. A projective unitary representation $σ$ of Möb is said to be associated with an operator T if $φ(T)= σ(φ)^\star T σ(φ)$ for all $φ$ in Möb. In this paper, we develop a Möbius equivariant version of the Sz.-Nagy--Foias model theory for completely non-unitary (cnu) contractions. As an application, we prove that if T is a cnu contraction with associated (projective unitary) representation $σ$, then there …
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…
