THE KLUVANEK-KANTOROVITZ CHARACTERIZATION OF SCALAR OPERATORS IN LOCALLY CONVEX SPACES

This paper is devoted to a proof of the characterization without duality theory, using strong integrals, while eliminating any assumptions of barrelledness or equicontinuity.

Perhaps the most useful aspect of our work here is the fact that we are able to construct the proofs almost entirely without duality theory.The use of linear functionals is kept to a minimum, and while some of the results (especially some lemmas) appear to depend on local convexity, this can even be avoided as in [1].
However, local convexity is apparently necessary to avoid certain problems with some of the integrals involved and to allow the use of results in [h].The reader W.V. SMITH will also note that the spaces involved are not assumed to be barrelled nor the spectral measures equicontinuous.This is an improvement over previous work in the area.
2. NOTATION AND PRELIMINARIES.L(X) will be the continuous endomorphisms of a quasicomplete separated locally convex topological vector space X; we assume L(X) is quasicomplete for simple con- S will be a locally compact Hausdorff space and C0(S) will be the con- vergence.
tinuous complex valued functions on S which vanish at infinity, llfl[ will denote the supremum norm of f, and will den'ote the Fourier transform of a function f on 3 CKARACTERIZATION.
THEOREM 3.1.Let T e L(X).T is scalar with real spectrum if and only if (+) {F f(s)eiSTxds IIII <_ i, f LI(-,)} is a relatively weally coml:.ctsubset of X for x e X.
PROOF OF TttNOREM 3.1.The map ' + f f(s )eSTds is continuous and weakly com- pact and, therefore, by Lemma 2.2, .f(s)eiSTxds I du x for each x e X.If x' e X' the continuous dual of X then recalling that la x is bounded ./edx(S) Ux(E).Then P is L(X) valued countably additive and regular (in the strong operator topology).Now it is easy to see that if *g(x) [ f(x-y)g(y)dy, then [ f*g(s)eiStxds f(t)eitTdt / g(u)eiUTxdu for all f, g LI(II).Therefore since f.g fg, (P (E) P(E)x) x f 9dP x f f dP x for all f g e LI(IRI) Since i is dense in C0(]RI), by Lemma 2.2, P is multi- plicative and setting t 0 in (3.1) gives P(II) I the identity in L(X).Hence P is a spectral measure.
ist i e Notice it s as t 0 and using Proposition (h.1) and Proposition (5.h) of [7] we have (the proof of (5.) is what is required) x W.V. SMITH for all x in a dense subset of X and if 8 [-n,n], then it is clear that TP (6 )x n n P(6 )Tx /-Tx for all x by continuity of T and since kdP is closed and n x -Tx we have -kdP Tx for all x.Therefore, T is scalar.lim dPP(Sn)X x n-isT isT e-iSdp and one can show that e x Conversely, if Tx kdPx, then e x x is continuous in s and bounded as in [2].Furthermore, f(s)eiSTxds exists as a Pettis integral by Thomas [hi for every f a LI(I), and this implies or ds / f(s)eiSTxds f(s)e-iSdp() x ( f(s)eiSTxds x') ( f(s)e-iSdp ()ds x') X f(s)e-iSd(Px(),x')ds fCs)e-iSdsd(P(X)x,X') f(s) eiSTxds ()dP x() and, therefore, (+) is relatively weakly com- and so pact by Lemma 2.2.This completes the proof.
A very well-known result of Bochner is: THEOREM 3.2.If (t) is continuous for < t < and has the property that _< K sup Cr e r r 1 zaR r 1 holds for all finite complex sequences {c and rational sequences {t then there r r exists a complex measure s.t.

$(t)
I eitZd(z), IIII < K Using this result and the methods of 3.1.above and Lemma 7 of [3], we obtain the follow,ing (we omit the proof).
THEOREM 3.3.Suppose the operator T e L(X) satisfies the following where x a A a bounded subset of X and x' a A' an equicontinuous subset of X' Suppose in addition that X is weakly sequentially complete.Then T is a scalar operator with real spectrum.