ON THE ESSENTIAL APPROXIMATE POINT SPECTRUM OF OPERATORS

If lambda belongs to the essential approximate point spectrum of a Banach space operator T is-an-element-of B(X) and {a(j)}j=0 infinity is a sequence of positive numbers with lim(j --> infinity) a(j) = 0, then there exists x is-an-element-of X such that \\p(T)x\\ greater-than-or-equal-to a(deg p) p(lambda)\ for every polynomial p. This result is the best possible - if \\p(T)x\\ greater-than-or-equal-to c\p(lambda)\ for some constant c > 0 then T has already a non-trivial invariant subspace, which is not true in general.