Convergence of orthogonal series of projections in Banach spaces

For a sequence ${\displaystyle \left(A_j\right)}$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums ${\displaystyle S_n={\sum }^n\_\{j=1\}{A}_j}$ 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 ${\displaystyle S_n}$ (i.e. ${\displaystyle S_{n}f\to Af}$ μ-a.e. for all f ∈ (A)).