Derivations and (hyper)invariant subspaces of a bounded operator

Shuang Zhang
1988 Proceedings of the American Mathematical Society  
Let X be a complex Banach space and £.{X) the set of bounded linear operators on X. For T 6 Z(X), a derivation At is defined by At A = TA -AT for A € £(X). By induction, A™ = AT o A™-1 is defined for each integer m > 2. We call the kernel of A™ the m-commutant of T. For a polynomially compact operator T, we consider the (hyper)invariant subspace problem for operators in the m-commutant of T for m > 1. It is easily seen that the m-commutant (m > 1) of T could be much larger than KerfAx). So our
more » ... an KerfAx). So our idea is a variation of Lomonosov's theorem in [6] . We start with several identities on derivations, and then prove our results on the existence of (hyper)invariant subspaces. Theorem 2 in [5] is generalized.
doi:10.1090/s0002-9939-1988-0920983-5 fatcat:ebak2p3lrveqdjxflvfzo6zppy