## Spectral projections.(English)Zbl 0556.47001

If $$\lambda\in {\mathbb{C}}$$ is not an accumulation point of the spectrum of a bounded linear operator S on a complex Banach space X then a familiar Cauchy integral gives rise to the ”spectral projection” Q($$\lambda$$,S) for the operator S at the point $$\lambda$$. This note is in response to the feeling that while it is all right to use such heavy machinery to construct the spectral projection, it ought to be possible in a much more simple way to define the projection $$P=I-Q(\lambda,S)$$ as a function of the operator $$T=S-\lambda I.$$ Our definition assumes that for example P is in the commutant of T, but enables us to prove that P is in the double commutant of T; if in particular T is a Fredholm operator then P is the product of T and its ”Drazin inverse”.

### MSC:

 47A10 Spectrum, resolvent 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.