In the Amemiya-Ando problem 3 is enough

TL;DR

This paper demonstrates that in infinite-dimensional Hilbert spaces, three specific orthogonal projections are sufficient to generate divergence in norm through sequences of their compositions, highlighting a fundamental property of such spaces.

Contribution

It proves that only three orthogonal projections are needed to produce divergence sequences in infinite-dimensional Hilbert spaces, simplifying previous understanding.

Findings

Existence of divergence sequences with three projections

Divergence occurs in infinite-dimensional Hilbert spaces

Sequences diverge in norm for some initial vectors

Abstract

In any infinite dimensional Hilbert space $\mathcal H$ there exist orthogonal projections $Q_1$, $Q_2$ and $Q_3$, such that a sequence $(P_n... P_1(x))$ diverges in norm for some $P_1,P_2,...\in\{Q_1,Q_2,Q_3\}$ and $x\in\mathcal H$.