A geometrical look at iterative methods for operators with fixed points

We distinguish classes of operators T with fixed points on a real Hilbert space by comparing the distances of a point x and its image Tx to the (set of) fixed points of T; this leads to a ranking of those classes, based on a nonnegative parameter. That same parameter also lets us conclude about the sign of and an upper bound for a characteristic inner product result that arises in iterative processes to obtain a common fixed point of a set of operators. We use that parameter (is the starting point for a geometrically-inclined study of specific iterative algorithms intended to find a common fixed point of operators belonging to such class.