Abstract
Simultaneous contractifications, simultaneous proper contractifications and semigroup (countable family or finite family) of commuting operators and of non-commuting perators are first given. Characterizations are given for a single bounded linear operator being a topological proper contraction. By using complexification of a real Banach space and by applying a fixed point theorem of Edelstein, it is shown that every compact topological strict contraction on a Banach space is a topological proper contraction. Finally, results on simultaneous proper contractification are applied to study the stability of a common fixed point of maps which are Frechet differentiable at that point.
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