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.
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 .
The operator is a projector which commutes with , and hence in the decomposition
both terms and are invariant subspaces of the operator .
- The spectrum of the restriction of to the subspace is contained in the region ;
- 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: