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.^[1]^[2]

Definition

Let $A$ be a closed linear operator in the Banach space ${\mathfrak {B}}$ . 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 by

P_{\Gamma }=-{\frac {1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1}\,\mathrm {d} z;

here $I_{\mathfrak {B}}$ is the identity operator in ${\mathfrak {B}}$ .

If $\lambda \in \sigma (A)$ is the only point of the spectrum of $A$ in $G_{\Gamma }$ , then $P_{\Gamma }$ is denoted by $P_{\lambda }$ .

Properties

The operator $P_{\Gamma }$ is a projector which commutes with $A$ , and hence in the decomposition

{\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 ${\mathfrak {L}}_{\Gamma }$ and ${\mathfrak {N}}_{\Gamma }$ are invariant subspaces of the operator $A$ . Moreover,

The spectrum of the restriction of $A$ to the subspace ${\mathfrak {L}}_{\Gamma }$ is contained in the region $G_{\Gamma }$ ;
The spectrum of the restriction of $A$ to the subspace ${\mathfrak {N}}_{\Gamma }$ 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.

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.

[1] Riesz, F.; Sz.-Nagy, B. (1956). Functional Analysis. Blackie & Son Limited.

[2] 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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v t e Spectral theory and ^*-algebras
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Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
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Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

Definition

Properties

See also

References