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Register a new userHomogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains
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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 pairs of Cowen-Douglas operators are similar to direct sums of certain basic pairs.
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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 operators in the Cowen-Douglas class of the Euclidean unit ball in $mathbb C^n$.
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.
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 $mathcal A({Omega})$, where $Omega subseteq mathbb C^m$ is a bounded domain, consisting of complex valued functions which are holomorphic on some open set $U$ containing $overline{Omega}$, the closure of $Omega$. The book, Hilbert Modules over function algebra, R. G. Douglas and V. I. Paulsen showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The book, Spectral decomposition of analytic sheaves, J. Eschmeier and M. Putinar and the book, Analytic Hilbert modules, X. Chen and K. Guo, provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.
We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $delta_i$-unstable for some $delta_i$. So we partially answer a problem posted by Mu~{n}oz-Occhetta-Sol{a} Conde. In particular, if $X$ is a generalized Grassmannian $mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ splits as a direct sum of line bundles. We improve the main theorem of Mu~{n}oz-Occhetta-Sol{a} Conde when $X$ is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-M{u}lich-Barth theorem on rational homogeneous spaces.
In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $mathcal{T}(L^{infty})$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic subset. Finally, we show that our results apply for the closure of a radical polynomial ideal $I$ whose zero variety satisfies the above conditions. A key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset generalizing the result of Beatrouss paper $L^p$-estimates for extensions of holomorphic functions.
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