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. It was introduced by Frigyes Riesz in 1912.

Definition

Let be a closed linear operator in the Banach space. Let be a simple or composite rectifiable contour, which encloses some region and lies entirely within the resolvent set of the operator. Assuming that the contour has a positive orientation with respect to the region, the Riesz projector corresponding to is defined by
here is the identity operator in.
If is the only point of the spectrum of in, then is denoted by.

Properties

The operator is a projector which commutes with, and hence in the decomposition
both terms and are invariant subspaces of the operator.
Moreover,
  1. The spectrum of the restriction of to the subspace is contained in the region ;
  2. The spectrum of the restriction of to the subspace lies outside the closure of.
If and are two different contours having the properties indicated above, and the regions
and have no points in common, then the projectors corresponding to them are mutually orthogonal: