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

[math]\displaystyle{ P_\Gamma=-\frac{1}{2\pi \mathrm{i}}\oint_\Gamma(A-z I_{\mathfrak{B}})^{-1}\,\mathrm{d}z; }[/math]

here [math]\displaystyle{ I_{\mathfrak{B}} }[/math] is the identity operator in [math]\displaystyle{ \mathfrak{B} }[/math].

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

Properties

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

[math]\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}, }[/math]

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

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

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

[math]\displaystyle{ P_{\Gamma_1}P_{\Gamma_2} = P_{\Gamma_2}P_{\Gamma_1}=0. }[/math]

See also

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