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Cohomological characterization of pseudoconvexity in a Banach space

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Abstract.

Let X be a Banach space with a countable unconditional basis (e.g., X=ℓ2), Ω⊂X open. We show that Ω is pseudoconvex if and only if for each affine complex line L in X the sheaf cohomology group H 1(Ω,I) vanishes, where I is the ideal sheaf of all holomorphic functions on Ω that vanish on Ω∩L. We also give an example that the condition H q(Ω,𝒪)=0 for all q≥1 unlike in finite dimensions does not imply the pseudoconvexity of Ω . Lastly, we prove an interpolation result.

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Authors and Affiliations

  1. Department of Mathematics, University of California at Riverside, Riverside, CA, 92521-0135, USA

    Imre Patyi

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  1. Imre Patyi
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Correspondence to Imre Patyi.

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Mathematics Subject Classification (2002): 32T05, 46G20.

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Patyi, I. Cohomological characterization of pseudoconvexity in a Banach space. Math. Z. 245, 371–386 (2003). https://doi.org/10.1007/s00209-003-0550-y

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  • DOI: https://doi.org/10.1007/s00209-003-0550-y

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