Riesz Projector
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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 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

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

References

  1. ^ Riesz, F.; Sz.-Nagy, B. (1956). Functional Analysis. Blackie & Son Limited.
  2. ^ Gohberg, I. C; Kre?n, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I.



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