Operators commuting with Boolean algebras of projections of finite multiplicity

Abstract

The algebra of operators commuting with a Boolean algebra of projections of finite uniform multiplicity (in the sense of Bade [3]) has been studied by Foguel [10] and the author [12]. The aim of the present paper is to show that most properties described in these two papers can be extended (sometimes, under additional conditions) to the algebra 6 of all the operators which commute with a complete countably decomposable Boolean algebra of projections containing no projections of infinite uniform multiplicity. It should be mentioned that one cannot get interesting general results in the case in which there are projections of infinite uniform multiplicity in the Boolean algebra of projections, since every operator on a Banach space commutes with the Boolean algebra of projections composed from the identities 0 and I. We shall start by proving that for every operator A E 6 there is a sequence of projections belonging to the above mentioned Boolean algebra of projections which increases to the identity and A multiplied by any element of this sequence is a spectral operator. Relying on this result we study the spectrum of operators of E and give a necessary and sufficient condition for such an operator to be spectral. In the following section we generalize Theorem 8 of [12] showing that in 6 a strong limit of spectral operators on a Hilbert space is spectral provided that they are of the same finite type and their resolutions of the identity are uniformly bounded. Adequate examples elucidate why we require the boundedness of the type of the spectral operators in most theorems.