Block diagonalization in Banach algebras

“Reduction" of linear operators is effected by commuting projections; the spectrum of the operator is then the union of the spectra of its range and null space restrictions. Disjointness of these partial spectra implies that the projection “double commutes" with the operator, which in turn can be recognised as a curious kind of “exactness". Variants of this exactness correspond to various kinds of disjointness between the partial spectra.