In a foundational paper Operators Possessing an Open Set of Eigenvalues written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space H possessing an open set W (in complex plane) of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that dimension of the eigenspace at each point w in W is 1, then the map f on W, sending w to ker(T-w), admits a non-zero holomorphic section, say S, and therefore defines a line bundle LT on W.It is well known that the curvature KL of a line bundle LT is a complete invariant for the line bundle LT . On the other hand, define

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Last updated: 23 Jun 2024