On the random product of orthogonal projections in Hilbert space (Q2753303)

Let \(H\) be a real Hilbert space, \(\{P_1, \dots, P_N\}\) be a ``pool'' of projections, \(r\) denote a finite sequence (word) of elements \(r_i\in\{1, \dots, N\}, x_0\in H\) and \(\{x_n\}\) be defined by iteration \(x_i=P_{r_i}x_{i-1}\). Then \(\{x_n\}\) converges weakly in \(H\), cf. \textit{I. Amemiya} and \textit{T. Ando} [Acta Sci. Math. 26, 239-244 (1965; Zbl 0143.16202)]. An interesting question is whether \(\{x_n\}\) converges strongly. NEWLINENEWLINENEWLINE\textit{I. Halperin} [Acta Sci. Math. 23, 96-99 (1962; Zbl 0143.16102)], by using the so-called Halperin bound, proved that the cyclic iteration of projections converges strongly. By the Halperin bound \(k(T)\) of a mapping \(T\) we mean the smallest constant \(k\geq 0\) such that \(\|x-Tx\|^2\leq k(\|x\|^2-\|Tx\|^2)\), \(x\in H\). NEWLINENEWLINENEWLINEBy presenting the quadratic form \(k(\|x\|^2-\|Tx\|^2)-\|x-Tx\|^2\), where \(T\) is a product of projections, as a sum of squares of the norms of linear combinations of the intermediate iterates and then by minimizing \(k(T)\), the authors approach their following conjecture: NEWLINENEWLINENEWLINEFor any mapping \(T\) in the multiplicative semigroup generated by \(N\) projections, \( k(T)\) is less than or equal to the binomial coefficient \(N\) choose \(N/2\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].