Riesz projector explained

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.^{[1] [2]}

Definition

Let

A be a closed linear operator in the Banach space ak{B} . 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 byP _{\Gamma =- (1)/(2 \pi i)} \oint _{\Gamma(A-z} I _{ak{B} })^\,\mathrmz;here I _{ak{B} } is the identity operator in ak{B} .

If

λ \in\sigma(A) is the only point of the spectrum of A in G _{\Gamma} , then P _{\Gamma} is denoted by P _{λ} .

Properties

The operator

P _{\Gamma} is a projector which commutes with A , and hence in the decompositionak{B}= ak{L} _{\Gamma ⊕ ak{N}} _{\Gamma ak{L}} _{\Gamma=P} _{\Gammaak{B}, ak{N}} _{\Gamma=(I} _{ak{B} }-P_\Gamma)\mathfrak,both terms ak{L} _{\Gamma} and ak{N} _{\Gamma} are invariant subspaces of the operator A .Moreover,

1. The spectrum of the restriction of

A to the subspace ak{L} _{\Gamma} is contained in the region G _{\Gamma} ;

1. The spectrum of the restriction of

A to the subspace ak{N} _{\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 _{\Gamma _{1}} and G _{\Gamma _{2}} have no points in common, then the projectors corresponding to them are mutually orthogonal:P _{\Gamma _{1}} P _{\Gamma _{2}} = P _{\Gamma _{2}} P _{\Gamma _{1}} =0 .

See also

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

Notes and References

1. Book: Functional Analysis. Riesz, F.. Sz.-Nagy, B.. Blackie & Son Limited. 1956.
2. Book: Gohberg, I. C. Kreĭn, M. G.. Introduction to the theory of linear nonselfadjoint operators. 1969. American Mathematical Society, Providence, R.I..