Infinite products of arbitrary operators and intersections of subspaces in Hilbert space (Q390780)

Let \(S_1,S_2,\dots, S_m\) be \(m\) closed linear subspaces of a Hilbert space \((H,\langle\cdot,\cdot\rangle)\) with induced norm \(\|\cdot\|\) and let \(S\) be their intersection. A theorem of \textit{I. Halperin} [Acta Sci. Math. 23, 96--99 (1962; Zbl 0143.16102)] shows that, for each \(x\in H\), we have NEWLINE\[NEWLINE\lim_{n\to\infty} \|(P_{S_m}, P_{S_{m-1}}\cdots p_{S_1})^nx- P_S x\|= 0,\tag{1}NEWLINE\]NEWLINE where \(P\) stands for the orthogonal projection onto the corresponding subspace. The convergence in (1) need not be uniform (on bounded sets) and may arbitrarily slow.NEWLINENEWLINE In this paper, the authors try to exclude the projection operators, replacing them with arbitrary, possibly nonlinear operators, but still obtaining the same conclusion as in Halperin's theorem, i.e., convergence to the intersection \(S\) without obligatory uniformity.