Some Algebraic Operators and the Invariant Subspace Problem

We consider the resolvent algebra , and Deddens' algebra . It is shown that both R (A) and B (A-I) possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R (A) whenever R (A) not equal {A}'. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R (A) when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R (A) contains properly {A}' whenever A has an eigenvalue other than zero.