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

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

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:

$P_{\Gamma _{1}}P_{\Gamma _{2}}=P_{\Gamma _{2}}P_{\Gamma _{1}}=0.$

See also

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