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Caltech

Analysis Seminar

Tuesday, November 21, 2017
3:00pm to 4:00pm
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Building 15, Room 104
Products of projections in Hilbert space: The interplay of slow convergence of cyclic projections and non-convergence of random projections
Eva Kopecka, Institut für Mathematik, Universitat Innsbruck,

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.

For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at [email protected].