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

$P_{\mathrm{\Gamma }}=-\frac{1}{2\pi \mathrm{i}}{\oint }_{\mathrm{\Gamma }}(A-z{I}_{\mathfrak{𝔅}}{)}^{-1}\mathrm{d}z;$

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 $G_{\mathrm{\Gamma }}$, then $P_{\mathrm{\Gamma }}$ is denoted by $P_{\lambda }$.

Properties

The operator $P_{\mathrm{\Gamma }}$ is a projector which commutes with $A$, and hence in the decomposition

$\mathfrak{𝔅}={\mathfrak{𝔏}}_{\mathrm{\Gamma }}\oplus {\mathfrak{𝔑}}_{\mathrm{\Gamma }}{\mathfrak{𝔏}}_{\mathrm{\Gamma }}=P_{\mathrm{\Gamma }}\mathfrak{𝔅},{\mathfrak{𝔑}}_{\mathrm{\Gamma }}=(I_{\mathfrak{𝔅}}-P_{\mathrm{\Gamma }})\mathfrak{𝔅},$

both terms ${\mathfrak{𝔏}}_{\mathrm{\Gamma }}$ and ${\mathfrak{𝔑}}_{\mathrm{\Gamma }}$ are invariant subspaces of the operator $A$. Moreover,

1. The spectrum of the restriction of $A$ to the subspace ${\mathfrak{𝔏}}_{\mathrm{\Gamma }}$ is contained in the region $G_{\mathrm{\Gamma }}$;
2. The spectrum of the restriction of $A$ to the subspace ${\mathfrak{𝔑}}_{\mathrm{\Gamma }}$ lies outside the closure of $G_{\mathrm{\Gamma }}$.

If ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are two different contours having the properties indicated above, and the regions $G_{{\mathrm{\Gamma }}_1}$ and $G_{{\mathrm{\Gamma }}_2}$ have no points in common, then the projectors corresponding to them are mutually orthogonal:

$P_{{\mathrm{\Gamma }}_1}{P}_{{\mathrm{\Gamma }}_2}=P_{{\mathrm{\Gamma }}_2}{P}_{{\mathrm{\Gamma }}_1}=0.$

See also

• Spectrum (functional analysis)
• Decomposition of spectrum (functional analysis)
• Spectrum of an operator
• Resolvent formalism
• Operator theory