A note on disjoint invariant subspaces (Q1098078)

The purpose of this note is to show that a nonzero subspace \({\mathcal B}\subset L(H)\) satisfying the property \((A^{\sim}_ n)\) for every positive integer n possesses many disjoint invariant subspaces, where L(H) denotes the algebra of all bounded linear operators on a separable Hilbert space H. It follows from this result that operators in the class \(A_{\aleph_ 0}\) and hence operators in the class (BCP) have disjoint invariant subspaces [cf. \textit{H. Bercovici}, Trans. Am. Math. Soc. 288, 139-146 (1985; Zbl 0569.47008) and \textit{H. Bercovici}, \textit{C. Foias} and \textit{C. Pearcy}, CBMS Regional Conf. Ser. Math. 56, 108 p. (1985; Zbl 0569.47007)].