Projectors on Convex Sets in Reflexive Banach Spaces

Abstract

252 Selfadjoint operators in Hilbert space can be synthetized out of orthogQ nal projectors by the process of for~ing the integrals of numerical functions with respect to an increasing one-parameter family of proje~ tors. To be viable such a mechanism-known as spectral synthesis-r~ quires from projectors a certain number of algebraic properties. Not long ago I have shown [7,8,91 that these properties subsist if the class of linear projectors is enlarged so as to include projectors on closed convex cones, conceived as nearest point mappings, and thus I was able to synthetize a new class of operators, mostly nonlinear. But then, having freed the spectral theory from its original confinement I was faced with the question of how far one can go on extending it. For instance, would it be valid in spaces other than Hilbert space? It is precisely to this question that I am ad,dressing myself in this paper, beginning with the study of projectors in reflexive Banach spaces. A first basic question is to decide what projectors on convex sets should be. Nearest point mappings certainly do not qualify, as they form an unruly class devoid of any algebraic structure, nor does any class of operators mapping the space into itself, since for these many of the required properties do not even make sense. This realized, one is led to the vie~ that projectors must be mappings, perhaps multi valued, acting from the dual into the space, view which in Hilbert space is thoroughly concealed by the standard identification of the space with its dual. At this stage a choice offers itself in a most natural way: The projector on a closed convex set K in a real reflexi ve Banach space X is the mapping P K : X*-+ ZX assigning to each x* E X* the set of points minimizing t IIx*1I 2 + t IIxll 2-(x*,x) over K. A series of familiar looking results soon brings out the certainty of being on the right track. So reassured, I have proceeded to investigate these new mathematical objects, not so much on their own right but rather as possible instruments for the spectral theory. My results are inconclu

Cite this paper

@inproceedings{Zarantonello2014ProjectorsOC, title={Projectors on Convex Sets in Reflexive Banach Spaces}, author={Eduardo H Zarantonello}, year={2014} }