Vol. 110, No. 2, 1984

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Interpolation of Banach spaces and negatively curved vector bundles

Richard Rochberg

Vol. 110 (1984), No. 2, 355–376

Let D be the open disk of the complex plane and T the unit circle. Let {Bei𝜃} be a family of Banach spaces parametrized by the points ei𝜃 of T. The fundamental construction in the theory of complex interpolation of Banach spaces produces from this data a family of Banach spaces {Bz} which is parametrized by the points z of D and which has the given {Bei𝜃} as boundary values. Basic facts about this construction are summarized in §2. B = zD{Bz} can be regarded as a complex vector bundle with base manifold D. In this paper we study the differential geometry of B and related vector bundles. We show relationships between interpolation theoretic inequalities for families of Banach spaces and the signs of certain curvatures of the associated vector bundles.

Mathematical Subject Classification 2000
Primary: 46M35
Secondary: 32L99, 46M20, 53C99, 55R99
Received: 11 May 1982
Published: 1 February 1984
Richard Rochberg
Department of Mathematics
Washington University in St. Louis
Campus Box 1146
One Brookings Dr
Saint Louis MO 63130-4899
United States