Spectral projections. (English) Zbl 0556.47001

Harte, Robin

Ir. Math. Soc. Newsl. 11, 10-15 (1984).

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.

Keywords:

accumulation point of the spectrum; Cauchy integral; spectral projection; double commutant; Fredholm operator; Drazin inverse