Formal Taylor series and complementary invariant subspaces

Abstract

A class of operators (which includes the unilateral weighted shifts and the noninvertible bilateral weighted shifts on Hilbert spaces), with the property that every element of the commutant of the operator is canonically associated to a formal Taylor series, is characterized. Let T T be one of such operators, then the following result is true: T T has no nontrivial complementary invariant subspaces, no roots and no logarithm. This result can be partially extended to the case when “commutant” is replaced by “double commutant", then: T T has no nontrivial complementary hyperinvariant subspaces.