A product of three projections

Eva Kopecká ¹   ; Vladimír Müller ²

¹ Department of Mathematics University of Innsbruck A-6020 Innsbruck, Austria
² Institute of Mathematics Academy of Sciences of the Czech Republic Žitná 25 CZ-11567 Praha, Czech Republic

Studia Mathematica, Tome 223 (2014) no. 2, pp. 175-186

Résumé

Let $X$ and $Y$ be two closed subspaces of a Hilbert space. If we send a point back and forth between them by orthogonal projections, the iterates converge to the projection of the point onto the intersection of $X$ and $Y$ by a theorem of von Neumann. Any sequence of orthoprojections of a point in a Hilbert space onto a finite family of closed subspaces converges weakly, according to Amemiya and Ando. The problem of norm convergence was open for a long time. Recently Adam Paszkiewicz constructed five subspaces of an infinite-dimensional Hilbert space and a sequence of projections on them which does not converge in norm. We construct three such subspaces, resolving the problem fully. As a corollary we observe that the Lipschitz constant of a certain Whitney-type extension does in general depend on the dimension of the underlying space.

Détail

DOI : 10.4064/sm223-2-4

Keywords: closed subspaces hilbert space send point back forth between orthogonal projections iterates converge projection point intersection theorem von neumann sequence orthoprojections point hilbert space finite family closed subspaces converges weakly according amemiya ando problem norm convergence long time recently adam paszkiewicz constructed five subspaces infinite dimensional hilbert space sequence projections which does converge norm construct three subspaces resolving problem fully corollary observe lipschitz constant certain whitney type extension does general depend dimension underlying space