Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ near $z_0$ is invertible). Consequently, there exists a finite-rank operator $P$, and for any $h \in C^1[0,1]$ and any $z$ near $z_0$, the following relationship holds: $$[(I-P_{z})^{-1}h](0)=\frac{P(h)(0)}{z-z_0}+ \text{ analytic terms.}$$ Is there an effective approach to compute the residue $P(h)(0)$ solely from the information provided by $P_z$ and $h$?
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$\begingroup$ There seem to be some assumptions missing. Under the given assumptions there doesn't need to exist such a projection $P$. In fact, $I-P_z$ need not even be invertible for $z$ close to $z_0$. (By the way, you probably want to replace $1$ with $\lambda$, but this doesn't fix the aforementioned issues.) $\endgroup$– Jochen GlueckCommented Aug 11, 2023 at 19:14
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$\begingroup$ From the way the question is written, it is not so clear what you are after. According to the question, $h$ is arbitrary, so it is not clear what information it could give you about the residue. If all you know is the approximate value of $z_0$, then you could use any of the usual iterative eigenvalue/eigenvector approximation methods. Also, as noted in the other comment, perhaps you meant to write $z_0$ for the eigenvalue, or set $z_0=1$. $\endgroup$– Igor KhavkineCommented Aug 12, 2023 at 12:02
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$\begingroup$ Sorry for being pedantic, but the edit still does not give the existence of the finite-rank map $P$, since $z \mapsto I - P_z$ could have an essential singularity at $z_0$. To get the map $P$ one also needs $I - P_{z_0}$ to be a Fredholm operator (or equivalently - as one has invertibility close to $z_0$ - a semi-Fredholm operator). $\endgroup$– Jochen GlueckCommented Aug 13, 2023 at 9:01
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