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 $\mathfrak$. 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
:$P_\Gamma=-\frac\oint_\Gamma(A-z I_)^\,\mathrmz;$
here $I_$ is the identity operator in $\mathfrak$.

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
:$\mathfrak=\mathfrak_\Gamma\oplus\mathfrak_\Gamma \qquad \mathfrak_\Gamma=P_\Gamma\mathfrak, \quad \mathfrak_\Gamma=(I_-P_\Gamma)\mathfrak,$
both terms $\mathfrak_\Gamma$ and $\mathfrak_\Gamma$ are invariant subspaces of the operator $A$.
Moreover,
# The spectrum of the restriction of $A$ to the subspace $\mathfrak_\Gamma$ is contained in the region $G_\Gamma$;
# The spectrum of the restriction of $A$ to the subspace $\mathfrak_\Gamma$ 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 $G_$
and $G_$ have no points in common, then the projectors corresponding to them are mutually orthogonal:
:$P_P_ = P_P_=0.$

See also

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

References

{{SpectralTheory

Spectral theory