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تسجيل مستخدم جديدHomogeneous 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 compo
sition 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
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