Resolvent spaces for algebraic operators and applications

For each element a in the Banach algebra A, we define the resolvent space Ra and completely characterize it whenever a is algebraic. In particular, we find elements a with Ra≠{a}′. Then we consider the Banach algebra of operators L(X), and show that RA 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 RA≠{A}′.