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 formula_1 be a closed linear operator in the Banach space formula_2. Let formula_3 be a simple or composite rectifiable contour, which encloses some region formula_4 and lies entirely within the resolvent set formula_5 (formula_6) of the operator formula_1. Assuming that the contour formula_3 has a positive orientation with respect to the region formula_4, the Riesz projector corresponding to formula_3 is defined by
here formula_12 is the identity operator in formula_2.
If formula_14 is the only point of the spectrum of formula_1 in formula_4, then formula_17 is denoted by formula_18.
Properties.
The operator formula_17 is a projector which commutes with formula_1, and hence in the decomposition
both terms formula_22 and formula_23 are invariant subspaces of the operator formula_1.
Moreover,
If formula_31 and formula_32 are two different contours having the properties indicated above, and the regions formula_33
and formula_34 have no points in common, then the projectors corresponding to them are mutually orthogonal: