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
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
![{\displaystyle P_{\Gamma }=-{\frac {1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1}\,\mathrm {d} z;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29f2e5eae6d388c91fb9fbbe6a55614edd4c4df5)
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
![{\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc50c04fc1d91eae25e58cfd923246a0bd9219e2)
both terms
and
are invariant subspaces of the operator
. Moreover,
- 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:
![{\displaystyle P_{\Gamma _{1}}P_{\Gamma _{2}}=P_{\Gamma _{2}}P_{\Gamma _{1}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f82a0508df4bbf070e8297521d56d5f58014af)
See also
- Spectrum (functional analysis)
- Decomposition of spectrum (functional analysis)
- Spectrum of an operator
- Resolvent formalism
- Operator theory
References
- ^ Riesz, F.; Sz.-Nagy, B. (1956). Functional Analysis. Blackie & Son Limited.
- ^ Gohberg, I. C; Kreĭn, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I.
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