Maximal, minimal, and primary invariant subspaces (Q5949553)

The author investigates the collection \(\text{Lat }T\) of all invariant subspaces for a given operator \(T\) on a complex Banach space \(X\). After introducing the class of subcritical operators the main theorem may be formulated as follows: If the iterates of \(T\) have norms of polynomials growth, \(T\) has range of finite codimension and a left inverse of subcritical class, then every maximal invariant subspace of \(T\) has codimension one.NEWLINENEWLINENEWLINEIn order to apply this theorem to Banach spaces of \(K\)-valued holomorphic functions on the unit disc \(\mathbb{D}\) where \(K\) is another Banach space, the concept of an admissible Banach space is introduced. If \(E\) is admissible the shift operator \(S_E\) and the backward shift operator \(B_E\) are bounded operators in \(E\). If \(K\) is finite-dimensional the above theorem shows that every maximal invariant subspace of \(S_E\) has codimension one, provided that the iterates of \(S_E\) have norms of polynomial growth and \(B_E\) is of subcritical class.NEWLINENEWLINENEWLINEThe obtained results are applied to many classical Banach spaces of holomorphic functions on \(\mathbb{D}\). Among others, Bergmann spaces and Hardy spaces are treated in detail.