Invariant Subspaces and Spectral Conditions on Operator Semigroups

Heydar Radjavi
1997 Banach Center Publications  
Introduction. Let H be a complex Hilbert space of finite or infinite dimension, and let E be a collection of bounded linear operators on H. We say E is reducible if there exists a subspace of H, closed by definition and different from the trivial subspaces {0} and H which is invariant under every member of E. We call E triangularizable if the set of invariant subspaces under E contains a maximal subspace chain. These questions have been studied extensively and the central problems in the
more » ... e-dimensional case, i.e., the invariant subspace problem and the transitive algebra problem are still unsolved for arbitrary operators on H [20, 21]. We are interested in the effect of certain spectral conditions on reducibility and triangularizability of a collection E. If f is any function, defined at least on all products of members of E, we say that f is permutable if for every integer k, every permutation τ , and all A 1 , . . . , A k in E. Special cases of interest for us are when f is the trace, the spectral radius, or the spectrum itself. Permutability of each of these three functions is easily seen to be necessary for triangularizability of a collection of operators at least on a finite-dimensional H. Since E will be a multiplicative semigroup in many of our considerations, let us observe that in this case the definition of permutability simplifies for the three functions just named. For example, it follows from the identity tr(AB) = tr(BA) and the decomposability of permutations into adjacent transpositions that trace is permutable on a semigroup E of trace-class operators if and only if tr(ABC) = tr(CBA) for all A, B, C in E. The same proof works for the spectral radius since r(AB) = r(BA) for all operators A and B. Although the spectrum does not satisfy σ(AB) = σ(BA) in infinite dimensions,
doi:10.4064/-38-1-287-296 fatcat:bdzahayvejfsngzpoyyjvavhki