Abstract: l Introduction* The present paper is concerned with conditions under which the quasi-nilpotent part of a spectral operator is actually nilpotent of some order k.As might be expected, the case of a spectral operator on a Hubert space has been settled longest.(See [4].)The case of a Banach space has been treated quite thoroughly by C. A. McCarthy [7] who showed that with a certain rate of growth condition on Q, the nilpotent part of the spectral operator T = S + Q r satisfies Q m+2 -0, where the m is a positive integer involved in the rate of growth condition.He alsσ discusses more special cases in which Q m+1 = 0 and provides examples to show that these exponents are the lowest possible in each case.The question of extending these results to general locally convex spaces could not even be formulated until a theory of spectral operators in these spaces had been devised The work of C. Ionescu Tulcea [5] having laid the foundations in this area, we may now attempt to solve the problem of generalizing McCarthy's results.It is shown below that his theorems, and indeed some part of the proofs, may be carried over to the locally convex case, with a suitable reformulation of some of the conditions and reworking of some of the supporting theory.The basic assumptions are as follows.E denotes a locally convex linear topological space over the field, C, of complex numbers.Moreover, E is assumed to be separated, barrelled and quasi-complete.The strong dual of E is denoted by E'.The space of continuous linear mappings of E into itself is jSf(E, E), which we shall always assume to be given the topology j?~h of uniform convergence on the bounded subsets of E. We denote the adjoint of T by ί Γ, for each Te^?(E, E).The resolvent set of T, res Γ, is a certain subset of C, the one-point compactification of C. Specifically, λ e res T provided there is a neighborhood V λ of λ in C and a function R τ with domain V λ Π C and range in £f(E, E) such that (a ) the set {R τ (z)x: 2 e F λ ΓΊ C} is a bounded subset of E for each x e E, and (b) R τ (z)(zl-T) = (zl-T)R τ (z) = I for all zeV λ f]C.The complement of res T, in C, is the spectrum of Γ, denoted sp T. If co g sp T, then sp T is compact in C and we say T is regular.We