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.^[1][2]

Definition

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

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

here ${\displaystyle I_{\mathfrak {B}}}$ is the identity operator in ${\displaystyle {\mathfrak {B}}}$.

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

Properties

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

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

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

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

If ${\displaystyle \Gamma _{1}}$ and ${\displaystyle \Gamma _{2}}$ are two different contours having the properties indicated above, and the regions ${\displaystyle G_{\Gamma _{1}}}$ and ${\displaystyle G_{\Gamma _{2}}}$ have no points in common, then the projectors corresponding to them are mutually orthogonal:

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

See also

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

References