A product of three projections

Eva Kopecká, and Vladimír Müller. "A product of three projections." Studia Mathematica 223.2 (2014): 175-186. <http://eudml.org/doc/285413>.

@article{EvaKopecká2014,
abstract = { 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. },
author = {Eva Kopecká, Vladimír Müller},
journal = {Studia Mathematica},
keywords = {Hilbert space; projection; product; extension},
language = {eng},
number = {2},
pages = {175-186},
title = {A product of three projections},
url = {http://eudml.org/doc/285413},
volume = {223},
year = {2014},
}

TY - JOUR
AU - Eva Kopecká
AU - Vladimír Müller
TI - A product of three projections
JO - Studia Mathematica
PY - 2014
VL - 223
IS - 2
SP - 175
EP - 186
AB - 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.
LA - eng
KW - Hilbert space; projection; product; extension
UR - http://eudml.org/doc/285413
ER -