Strange products of projections

TL;DR

This paper demonstrates that in infinite-dimensional Hilbert spaces, it is possible to construct three orthogonal projections onto closed subspaces such that certain iterative sequences do not converge in norm, revealing complex behaviors of projection sequences.

Contribution

The paper constructs specific three-projection sequences in infinite-dimensional Hilbert spaces that exhibit non-convergent behavior for all non-zero initial vectors.

Findings

Existence of three projections causing non-convergent sequences

Non-convergence occurs for all non-zero initial vectors

Highlights complex dynamics of projection sequences in infinite dimensions

Abstract

Let $H$ be an infinite dimensional Hilbert space. We show that there exist three orthogonal projections $X_1, X_2, X_3$ onto closed subspaces of $H$ such that for every $0\ne z_0\in H$ there exist $k_1, k_2,\dots \in \{1,2,3\}$ so that the sequence of iterates defined by $z_n= X_{k_n} z_{n-1}$ does not converge in norm.