Fredholm Alternative and Riesz Schauder Theorem

Theorem (Analytic Fredholm Alternative) Let {D} be a connected open subset of {{\mathbb C}} and {\mathcal{H}} be a separable Hilbert space. Suppose that {f : D \longrightarrow \mathcal{L} (\mathcal{H})} is an analytic operator-valued function such that {f(z)} is compact {\forall z \in D}. Then either

  1. {(I - f(z)) ^{-1}} does not exist {\forall z \in D}.
  2. {(I - f(z)) ^{-1}} exists {\forall z \in D\setminus S}, where {S} is discrete.

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