Commuting Boolean Algebras of Projections. II. Boundedness In L p

C. A. McCarthy
1964 Proceedings of the American Mathematical Society  
This paper complements our previous paper of the same title [4] . The conclusions, results applicable to the theory of spectral operators, are those of the previous paper, but the hypotheses are disjoint and the methods are somewhat different; we will make only historical reference to this work. One of the basic problems of the theory of spectral operators is whether the sum and product of two commuting spectral operators on a Banach space is again spectral (for background material on spectral
more » ... perators see especially [l] or [2]). Wermer [5] showed that this is in fact always the case if the operators act on a Hilbert space. Dunford [l] and Foguel [3] proved that if the underlying space is weakly complete, the boundedness of the Boolean algebra of projections generated by the resolutions of the identity of the operators implies that the sum and product is spectral. In practice, however, this may be difficult to determine, and our work has been to find easily applicable criteria for this boundedness. Our previous paper [4] gave a criterion in terms of multiplicity: It sufficed that one of the algebras of projections was of finite multiplicity, and that even for some separable reflexive Banach spaces this condition was necessary. Our present criterion is in terms of the underlying space and independent of various properties the generating Boolean algebras might enjoy other than their boundedness. Our result holds for Lp spaces, 1 <p < oo, direct sums of Lv spaces for which the p's are bounded away from 1 and oo, and for subspaces thereof; especially of interest for partial differential equations is that our theorem holds for the Sobolev spaces where the norm of a function x is given by the sum of the Lp norm of x and of some perhaps different Lp norms of its derivatives. Since our estimates are all finite-combinatoric, our results hold for inseparable Lp spaces, L" spaces with respect to only finitely additive measures, and their elaborations. Our first section will establish some combinatorial propositions, which when used in the second section will give our theorem for 2 Sp < °° ; consideration of adjoints gives the theorem for 1 <p S2.
doi:10.2307/2034597 fatcat:mksoxokmrbbsthx3qam6evkqq4