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I'm wondering whether there is such a thing as a "residue theorem for holomorphic operator-valued functions". More precisely, I want to evaluate an integral of the form

$P:=\int_{\Gamma} (A(\lambda) - \lambda)^{-1} d \lambda$

where each $A(\lambda)$ is a closed operator and $\Gamma$ encloses an eigenvalue $\lambda_0$ of the holomorphic operator pencil $A(\lambda) - \lambda$, i.e., for some eigenfunction $u_0$ we have $A(\lambda_0)u_0 - \lambda_0u_0=0$.

In the case of a $\lambda$-independent $A$ the operator $P$ is (up to a constant) the well-known Riesz Projection corresponding to $\lambda_0$. If (and how) this can be generalized to a $\lambda$-nonlinear eigenvalue problem is precisely what my question is concerned with.

Thanks for any help in advance!

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+1 cool question –  draks ... Mar 14 '12 at 18:47

1 Answer 1

up vote 4 down vote accepted

Beyond the fairly standard discussion of resolvents, there is a reasonable "Cauchy theory" for vector-valued holomorphic and meromorphic functions, with values in a quasi-complete locally convex topological vector space. Rudin's Functional Analysis discusses the Frechet-space-valued case fairly thoroughly, with some abstractions. My functional analysis notes at http://www.math.umn.edu/~garrett/m/fun/ include discussion of quasi-completeness, weak-and-strong holomorphy, etc. Bourbaki's "Integration" talks about vector-valued integrals in this generality, too.

The potentially delicate point is what "meromorphy" means for a vector-valued function $F$: basically, there must be a scalar-valued holomorphic function $f$ so that $f\cdot F$ is holomorphic. That is, one must avoid bad singularities. Granting that, the residue calculus works as well as one could reasonably hope!

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Thanks very much, that is of great help to me! –  user26895 Mar 15 '12 at 8:49
@user26895 Very good. :) –  paul garrett Mar 15 '12 at 12:36

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