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

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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

 

here   is the identity operator in  .

If   is the only point of the spectrum of   in  , then   is denoted by  .

Properties

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The operator   is a projector which commutes with  , and hence in the decomposition

 

both terms   and   are invariant subspaces of the operator  . Moreover,

  1. The spectrum of the restriction of   to the subspace   is contained in the region  ;
  2. 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:

 

See also

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References

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  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.