Riesz projector

In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912.

Definition
Let $$A$$ be a closed linear operator in the Banach space $$\mathfrak{B}$$. Let $$\Gamma$$ be a simple or composite rectifiable contour, which encloses some region $$G_\Gamma$$ and lies entirely within the resolvent set $$\rho(A)$$ ($$\Gamma\subset\rho(A)$$) of the operator $$A$$. Assuming that the contour $$\Gamma$$ has a positive orientation with respect to the region $$G_\Gamma$$, the Riesz projector corresponding to $$\Gamma$$ is defined by

P_\Gamma=-\frac{1}{2\pi \mathrm{i}}\oint_\Gamma(A-z I_{\mathfrak{B}})^{-1}\,\mathrm{d}z; $$ here $$I_{\mathfrak{B}}$$ is the identity operator in $$\mathfrak{B}$$.

If $$\lambda\in\sigma(A)$$ is the only point of the spectrum of $$A$$ in $$G_\Gamma$$, then $$P_\Gamma$$ is denoted by $$P_\lambda$$.

Properties
The operator $$P_\Gamma$$ is a projector which commutes with $$A$$, and hence in the decomposition
 * $$\mathfrak{B}=\mathfrak{L}_\Gamma\oplus\mathfrak{N}_\Gamma

\qquad \mathfrak{L}_\Gamma=P_\Gamma\mathfrak{B}, \quad \mathfrak{N}_\Gamma=(I_{\mathfrak{B}}-P_\Gamma)\mathfrak{B}, $$ both terms $$\mathfrak{L}_\Gamma$$ and $$\mathfrak{N}_\Gamma$$ are invariant subspaces of the operator $$A$$. Moreover,
 * 1) The spectrum of the restriction of $$A$$ to the subspace $$\mathfrak{L}_\Gamma$$ is contained in the region $$G_\Gamma$$;
 * 2) The spectrum of the restriction of $$A$$ to the subspace $$\mathfrak{N}_\Gamma$$ lies outside the closure of $$G_\Gamma$$.

If $$\Gamma_1$$ and $$\Gamma_2$$ are two different contours having the properties indicated above, and the regions $$G_{\Gamma_1}$$ and $$G_{\Gamma_2}$$ have no points in common, then the projectors corresponding to them are mutually orthogonal:

P_{\Gamma_1}P_{\Gamma_2} = P_{\Gamma_2}P_{\Gamma_1}=0. $$