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 {\displaystyle A} be a closed linear operator in the Banach space B {\displaystyle {\mathfrak {B}}} . Let Γ {\displaystyle \Gamma } be a simple or composite rectifiable contour, which encloses some region G Γ {\displaystyle G_{\Gamma }} and lies entirely within the resolvent set ρ ( A ) {\displaystyle \rho (A)} ( Γ ⊂ ρ ( A ) {\displaystyle \Gamma \subset \rho (A)} ) of the operator A {\displaystyle A} . Assuming that the contour Γ {\displaystyle \Gamma } has a positive orientation with respect to the region G Γ {\displaystyle G_{\Gamma }} , the Riesz projector corresponding to Γ {\displaystyle \Gamma } is defined by
P Γ = − 1 2 π i ∮ Γ ( A − z I B ) − 1 d z ; {\displaystyle P_{\Gamma }=-{\frac {1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1}\,\mathrm {d} z;}
here I B {\displaystyle I_{\mathfrak {B}}} is the identity operator in B {\displaystyle {\mathfrak {B}}} . If λ ∈ σ ( A ) {\displaystyle \lambda \in \sigma (A)} is the only point of the spectrum of A {\displaystyle A} in G Γ {\displaystyle G_{\Gamma }} , then P Γ {\displaystyle P_{\Gamma }} is denoted by P λ {\displaystyle P_{\lambda }} .
Properties The operator P Γ {\displaystyle P_{\Gamma }} is a projector which commutes with A {\displaystyle A} , and hence in the decomposition
B = L Γ ⊕ N Γ L Γ = P Γ B , N Γ = ( I B − P Γ ) B , {\displaystyle {\mathfrak {B}}={\mathfrak {L}}_{\Gamma }\oplus {\mathfrak {N}}_{\Gamma }\qquad {\mathfrak {L}}_{\Gamma }=P_{\Gamma }{\mathfrak {B}},\quad {\mathfrak {N}}_{\Gamma }=(I_{\mathfrak {B}}-P_{\Gamma }){\mathfrak {B}},}
both terms L Γ {\displaystyle {\mathfrak {L}}_{\Gamma }} and N Γ {\displaystyle {\mathfrak {N}}_{\Gamma }} are invariant subspaces of the operator A {\displaystyle A} . Moreover,
… excerpt ends here. Continue reading the full article.
