[Paper Review] The Amemiya-Ando conjecture falls

Adam Paszkiewicz|arXiv (Cornell University)|Mar 15, 2012

Optimization and Variational Analysis3 references9 citations

TL;DR

This paper resolves the Amemiya-Ando conjecture by proving that in any infinite-dimensional Hilbert space, the product of projections from a finite set of five orthogonal projections can diverge in norm for some vector. The proof constructs a sequence of projections using unitary rotations and iterative approximation to show norm divergence, thereby disproving the long-standing conjecture that such products always converge strongly.

ABSTRACT

In any infinite dimensional Hilbert space H, a sequence P_n...P_1 x diverges in norm for some x \in H and orthogonal projections P_n \in {Q_1,..., Q_5}.

Motivation & Objective

• To resolve the Amemiya-Ando conjecture, which posits that products of projections from a finite set of projections in a Hilbert space converge strongly.
• To demonstrate that norm convergence does not hold in general for products of more than two projections, even when the projections are orthogonal and finite in number.
• To construct explicit counterexamples in infinite-dimensional Hilbert spaces where the product of projections diverges in norm for some initial vector.
• To establish that the conjecture fails even with only five projections, thereby closing a long-open problem in operator theory.

Proposed method

• Constructs a sequence of nested projections $ P_0 \leq \dots \leq P_r $ with $ \dim(P_s - P_{s-1}) = 1 $ using unitary transformations on auxiliary vectors orthogonal to a fixed 2D subspace.
• Employs unitary operators $ U_s $ that rotate vectors in the orthogonal complement of a fixed 2D projection $ E = \hat{e} + \hat{e}' $, enabling control over the limit behavior of iterated projections.
• Uses the key identity $ (E U_s P U_s^* E)^n \to \hat{f}_s $ as $ n \to \infty $ and $ \delta_s \to 0 $, allowing approximation of any one-dimensional projection within $ E $.
• Applies Lemma 1 to approximate any finite sequence of one-dimensional projections within a 2D subspace using products of projections from a larger set.
• Constructs a global counterexample by tiling the Hilbert space with orthogonal subspaces $ F_i $, each supporting a monomial that amplifies a component in a specific direction.
• Combines alternating monomials $ A_{2k}(P,Q,E) $ and $ A_{2k+1}(R,S,E) $ across disjoint subspaces to generate a vector sequence with bounded, non-converging components.

Experimental results

Research questions

• RQ1 Does the product of projections from a finite set of orthogonal projections in an infinite-dimensional Hilbert space always converge in norm?
• RQ2 Can a sequence of products of projections from five orthogonal projections diverge in norm for some initial vector?
• RQ3 Is the Amemiya-Ando conjecture, which suggests strong convergence of such products, valid in infinite-dimensional Hilbert spaces?
• RQ4 What is the minimal number of projections required to achieve norm divergence in infinite-dimensional Hilbert space?

Key findings

• The Amemiya-Ando conjecture is false: in any infinite-dimensional Hilbert space, there exist five orthogonal projections $ Q_1, \dots, Q_5 $ such that $ P_n \dots P_1 x $ diverges in norm for some $ x \in H $.
• For any $ \epsilon > 0 $, there exist projections $ P_0 \leq \dots \leq P_r $ with $ \dim(P_s - P_{s-1}) = 1 $ such that $ \| (E P_s E)^{n(s)} - \hat{f}_s \| < \epsilon $, allowing approximation of any finite sequence of one-dimensional projections in a 2D subspace.
• The construction ensures that $ \| (E (P Q P)^{m(s)} E)^{n(s)} - \hat{f}_s \| < \epsilon $, proving that monomials from three projections can simulate arbitrary one-dimensional projections in a 2D subspace.
• By tiling the space with orthogonal subspaces $ F_i $, each supporting a monomial that amplifies a component by a factor $ \eta_i > 1 - \epsilon_i $, the product sequence accumulates non-vanishing components, leading to norm divergence.
• The product sequence $ A_{2k}(P,Q,E) A_{2k-1}(R,S,E) \dots A_1(R,S,E) e_1 $ has components $ \langle x_{2k}, e_{2k+1} \rangle = \eta_1 \dots \eta_{2k} $, and since $ \prod \eta_i > 0 $, the sequence does not converge in norm.
• The counterexample uses only five projections: $ \{P, Q, R, S, E\} $, where $ E = \sum \hat{e}_i $, showing that five projections suffice to break norm convergence.