Shifts on Hilbert spaces.

Does every operator on an infinite-dimensional Hubert space have a non-trivial invariant subspace ? The question is still unanswered. A possible approach is to classify all invariant subspaces of all known operators in the hope of getting an insight that will lead to a proof or a counterexample; this paper is a step in that direction. The operators selected for study are certain special isometries, called shifts. There are two reasons for this choice. First, shifts constitute perhaps the simplest and most natural class of operators for which the classification problem is solvable but not trivial. Second, shifts are known to be typically infinite-dimensional operators, and it is not unreasonable to hope that their study, even if it does not lead to a solution of the existence problem, might make further work on the classification problem easier.