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Nonexpansive retracts in Banach spaces

100%

Kopecká E. , Reich S.

Banach Center Publications

2007

tom 77

nr 1

161-174

We study various aspects of nonexpansive retracts and retractions in certain Banach and metric spaces, with special emphasis on the compact nonexpansive envelope property.

A product of three projections

100%

Kopecká E. , Müller V.

Studia Mathematica

2014

tom 223

nr 2

175-186

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.