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 not equal {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 not equal {A}'. (C) 2013 Published by Elsevier Inc.