M.Sc Thesis
M.Sc Student Salinas Pillajo Zuly Leonela Infinite Products of Operators Department of Mathematics Professor Emeritus Simeon Reich

Abstract

Given a finite number of closed and convex subsets of certain non-Hilbert spaces, the intersection of which is nonempty, we prove the convergence, either strong or weak, of methods for finding a point in that intersection. These methods involve possibly discontinuous operators as well as their infinite products or infinite products of their convex combinations. The problem of finding a point in the intersection of convex and closed sets is referred to as the convex feasibility problem and has applications, for example, in the image recovery field. One of the first algorithms to solve it was proposed by J. von Neumann in the early 1930s, solving the problem for the intersection of two closed subspaces in a Hilbert space. Years later, I. Halperin extended von Neumann's idea to the intersection of a finite number of subspaces. Since then, the interest in this problem has increased and as result, it has been extended to more general cases, for instance, outside Hilbert spaces, nonlinear cases, weak versions, etc.

In this connection, E. Pustylnik and S. Reich have recently proved the following result. Consider the orthogonal projections  of a Hilbert space  onto closed subspaces . Consider also the possibly nonlinear operators , ; , and suppose that for all , the inequalities

hold for some positive numbers  with . Then, for each , there exists a point  such that

Our aim is to extend Pustylnik's and Reich's result to possibly discontinuous operators defined outside Hilbert spaces; more precisely, on Banach spaces, the Hilbert ball  and CAT(0) spaces. To this purpose we use norm-one projections, retractions or nearest point projections instead of orthogonal projections. These operators are then approximated by other, possibly nonlinear and even discontinuous operators.