L-p-projections on subspaces and quotients of Banach spaces

摘要

The aim of this paper is to study L-p-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An L-p- projection on a Banach space X, for 1 <= p <= + infinity, is an idempotent operator P satisfying vertical bar vertical bar f vertical bar vertical bar(x) = vertical bar vertical bar(vertical bar vertical bar P(f)vertical bar vertical bar(x), vertical bar vertical bar(I-P)(f)vertical bar vertical bar(x))vertical bar vertical bar l(p) for all f is an element of X. This is an Lp version of the equality vertical bar vertical bar f vertical bar vertical bar(2) = vertical bar vertical bar Q(f)vertical bar vertical bar(2) + vertical bar vertical bar(I-Q) (f)vertical bar vertical bar(2), valid for orthogonal projections on Hilbert spaces. We study the relationships between L-p- projections on a Banach space X and those on a subspace F, as well as relationships between L-p- projections on X and those on the quotient space X/F. All the results in this paper are true for 1