Holomorphic Spectral Theory: A Geometric Approach

The paper introduces geometric, analytic, and operator theoretic concepts in holomorphic spectral theory, the part of spectral theory developed by using holomorphic operator valued mappings and differential forms. The objects of interest analyzed in this paper are holomorphic mappings with values in Grassmann manifolds of Hilbert spaces, Hermitian holomorphic vector bundles, and Cowen–Douglas classes of operators. A related goal is to identify complete sets of invariants for these objects. As specific issues in the geometric approach to holomorphic spectral theory, one should mention the Cauchy–Riemann equations for holomorphic families of closed subspaces of a Hilbert space, the Gram–Schmidt operator that associates a projection to each idempotent in algebras of Hilbert space operators, and an operator theoretic substitute for the generalized Grothendieck lemma in complex analysis.