Convergence of orthogonal series of projections in Banach spaces
Volume 66 / 1997
Annales Polonici Mathematici 66 (1997), 137-153
DOI: 10.4064/ap-66-1-137-153
Abstract
For a sequence $(A_j)$ of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums $S_n = ∑^n_{j=1} A_j$ 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 $S_n$ (i.e. $S_{n}f → Af$ μ-a.e. for all f ∈
