Infinite products of arbitrary operators and intersections of subspaces in Hilbert space

Let mm closed linear subspaces S1,…,SmS1,…,Sm of a given Hilbert space HH and mm sequences of arbitrary operators (An(k))n=1∞,k=1,…,m, be connected by the relations limn→∞‖An(k)x−PSkx‖=0 for each x∈H,k=1,…,m, where PP denotes the orthogonal projection onto the corresponding subspace. We derive sufficient conditions on the operators An(k), which yield strong convergence of the infinite products ∏n=1∞(An(m)⋯An(1))x for any x∈Hx∈H, with the limits belonging to the intersection of all the subspaces S1,…,SmS1,…,Sm. Several counterexamples show the optimality of our conditions.