Projectivity and Spectrality

Spectral Decomposition with Respect to a Circle

Consider that \( T \) is an endomorphism of a finite-dimensional vector space \( E \), and let \( S \) be a circle in the complex plane which does not pass through any eigenvalue of \( T \). Then we consider the operator:
\[\mathrm{Q}=\frac{1}{2 \pi i} \int_{S}(z-T)^{-1} d z\]
$\mathrm Q$ is a projection operator in \( E \) which commutes with \( T . \) The decomposition
\[E=E_{+} \oplus E_{-}, \quad E_{+}=Q E, \quad E_{-}=(1-Q) E\]
is therefore invariant under \( T \), so that we can write:
\[T=T_{+} \oplus T_{-}\]
Then \( T_{+} \) has all eigenvalues inside \( S \) while \( T_{-} \) has all eigenvalues outside \( S \). This is just the spectral decomposition of \( T \) corresponding to the two components of the complement of the circle \( S \).

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