A product of three projections

• Autorzy: Eva Kopecká , Vladimír Müller
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 54C20: Extension of maps
• 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2014
• Tom: 223
• Numer: 2
• Strony: 175-186

Daty

wydano

2014

Twórcy

autor

Eva Kopecká

• Department of Mathematics, University of Innsbruck, A-6020 Innsbruck, Austria

autor

Vladimír Müller

• Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, CZ-11567 Praha, Czech Republic

Identyfikatory

DOI

10.4064/sm223-2-4