Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T, that is, for the convergence of the sequence ${\left(Tⁿ\right)}_{n\in \mathbb{N}}$ of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ ℕ, the restriction of the pth iterate of T to the range of I-T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the Banach Contraction...

For a sequence $\left(A_j\right)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n={\sum }_{j=1}^n{A}_j$ in a “strong” sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of $S_n$ (i.e. $S_{n}f\to Af$ μ-a.e. for all f ∈ (A)).