Resolvent spaces for algebraic operators and applications

For each element a in the Banach algebra A , we define the resolvent space R a and completely characterize it whenever a is algebraic. In particular, we find elements a with R a ≠ { a } ′ . Then we consider the Banach algebra of operators L ( X ) , and show that R A possesses nontrivial invariant subspaces whenever A is an algebraic element of L ( X ) . This assertion becomes stronger than that of the existence of a hyper-invariant subspace for A whenever R A ≠ { A } ′ .