Spans and intersections of essentially reducing subspaces

Michael J. Hoffman
1978 Proceedings of the American Mathematical Society  
If P and Q are the projections onto essentially reducing subspaces M and N for an operator, the closed linear span and the intersection of M and N need not be essentially reducing or even essentially invariant. However, they are if M + N is closed, in particular if PQ = QP or if PQ is compact. 1. Introduction. If F is the orthogonal projection of separable Hilbert space 77 onto a closed subspace M, then M is invariant under a bounded linear operator T when TP -PTP = 0. It reduces F when TP -PT
more » ... 0. This paper examines these notions modulo the compact operators. Let L(77) be the algebra of all bounded linear operators on 77, and K(H) be the ideal of compact operators. Definition. If S ç L(77) and F is the orthogonal projection onto a closed subspace M, then M (or F) is essentially invariant for S if TP -PTP is in K(H) for every T in S. It is essentially reducing if TP -PT is in K(H) for every T in S. In contrast to the situation for invariant subspaces, every operator is known to have nontrivial (i.e. infinite dimension and codimension) essentially invariant and reducing subspaces. Fillmore, Stampfli, and Williams show that if c is in the boundary of the essential spectrum of T, there is a projection F of infinite rank and nullity such that (T -c)P is compact, so (1 -P)TP E K(H) [4, p. 815]. Recent results of Voiculescu show that separable algebras of operators have nontrivial essentially invariant projections [1, p. 344], [7] . In particular, the C*-algebra generated by T and 1 does, so F has essentially reducing subspaces. The examples in §2 show that much of the lattice structure of the sets of invariant and reducing subspaces for an operator fails for these weaker notions. We do have the following result. Proposition 1.1. Suppose M and N are closed subspaces of Hilbert space H and that M + N is closed. If M and N are essentially invariant (reducing) for an operator T, then M + N and M n N are also.
doi:10.1090/s0002-9939-1978-0507334-9 fatcat:d7aycqkb35gljm6tnaobjgckha