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Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spetrum 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 $h$?

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 $h$?

Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and $P_{z_0}$ possesses an eigenvalue $\lambda$ (assuming multiplicity is 1). 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 $h$?

Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spetrum 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 $h$?