Analysis Seminar

Let X and Y be two closed subspaces of a Hilbert space. If we send a point back and forth between them by orthogonal projection, the iterates converge to the projection of the point on the intersection of X and Y. Already on three subspaces X, Y and Z we can project either cyclically as above: X,Y,Z,X,Y,Z,X,Y,Z,... , or "randomly", for example: X,Y,X,Y,Z,Y,X,Y,Z,Y,Z,....

It turns out that these two cases possibly result in a completely different (non-)convergence behavior.

If the Hilbert space is finite dimensional, in both cases there is convergence [classical].

In infinite dimensional Hilbert spaces the cyclic products always converge [classical].

If, however, X,Y, and Z "almost touch", the cyclic product converges arbitrarily slowly, and a "random" product can diverge [recent].

It is not known, if "most" of the products converge when we fix the spaces X,Y,Z and a starting point.