Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections $$P_{\{\lambda_1,...\lambda_n\}}=\oint_\gamma(T-z)^{-1}dz,$$ where $\gamma$ is a smooth curve encircling $\{\lambda_1,...\lambda_n\} $ (and no other eigenvalues). One can show that $P^2=P$.

My Question:

Formally, one can split the curve $\gamma$ into $n$ curves encircling only $\lambda_i$, respectively. This suggests that $$P_{\{\lambda_1,...\lambda_n\}}=\sum_{i=1}^nP_{\{\lambda_i\}}.$$ Is this equation true?

If not, is there any sensible relationship between $P_{\{\lambda_1,...\lambda_n\}}$and $\sum_{i=1}^nP_{\{\lambda_i\}}$?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections $$P_{\{\lambda_1,...\lambda_n\}}=\frac{1}{2\pi i}\oint_\gamma(T-z)^{-1}dz,$$ where $\gamma$ is a smooth curve encircling $\{\lambda_1,...\lambda_n\} $ (and no other eigenvalues). One can show that $P^2=P$.

My Question:

Formally, one can split the curve $\gamma$ into $n$ curves encircling only $\lambda_i$, respectively. This suggests that $$P_{\{\lambda_1,...\lambda_n\}}=\sum_{i=1}^nP_{\{\lambda_i\}}.$$ Is this equation true?

If not, is there any sensible relationship between $P_{\{\lambda_1,...\lambda_n\}}$and $\sum_{i=1}^nP_{\{\lambda_i\}}$?