Differentiable projections and differentiable semigroups

Abstract

Suppose X is a Banach space, G is a connected open subset of X, and p is a continuously Fr6chet differentiable function from G into G satisfying p(p(x))=p(x) for each x in G. We prove that p(G) is a differentiable submanifold of X and use this result to show that the maximal subgroup containing an idempotent in a differentiable semigroup is a Lie group. Suppose X is a Banach space, G is a connected open subset of X, and p is a continuously Frechet differentiable function from G into G satisfying p(p(x))=p(x) for each x in G. We prove that p(G) is a differentiable submanifold of X and use this result to show that the maximal subgroup containing an idempotent in a differentiable semigroup is a Lie group. In [4] Nadler proved that the image of a differentiable projection defined on an open subset of a Euclidean space is a differentiable manifold. THEOREM 1. Suppose each of X, p, and G is as above. There is a closed linear subspace Y of X and a set H= {h.} offunctions, one for each x in p(G) so that, for each x in p(G), h_ is a homeomorphism from a neighborhood of x in p(G) onto a neighborhood of 0 in Y, h,(x)=0, and if each of x and y is in p(G) with y in dom(h_) then there is a neighborhood Ux, of 0 in Y so that hx o h71 is continuously differentiable on U,,,. First, we establish some notation. Since p2=p we have, by the chain rule, that p'(p(x)) o p'(x)=p'(x) for each x in G. In particular, if x is in p(G) then p'(x) is a continuous linear idempotent mapping from X into X. Denote by Yx the image of p'(x) if x is in p(G). Let px denote the function defined from G-x to G-x by p_(y)=p(y+x)-x for each x in p(G). Finally, if d is a positive number denote by R(d) the subset of X to which x belongs if and only if llxll O so that p'(x) is one-to-one on R(d,) npx(G-x). Received by the editors January 18, 1973 and, in revised form, February 26, 1973. AMS (MOS) subject classfications (1970). Primary 52B99; Secondary 22E65.