# Convergence of orthogonal series of projections in Banach spaces

• Volume: 66, Issue: 1, page 137-153
• ISSN: 0066-2216

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## Abstract

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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)).

## How to cite

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Ryszard Jajte, and Adam Paszkiewicz. "Convergence of orthogonal series of projections in Banach spaces." Annales Polonici Mathematici 66.1 (1997): 137-153. <http://eudml.org/doc/269982>.

@article{RyszardJajte1997,
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 ∈ (A)).},
author = {Ryszard Jajte, Adam Paszkiewicz},
journal = {Annales Polonici Mathematici},
keywords = {idempotent; mutually orthogonal projections; L₂-space; convergence almost everywhere; -space; idempotent operators},
language = {eng},
number = {1},
pages = {137-153},
title = {Convergence of orthogonal series of projections in Banach spaces},
url = {http://eudml.org/doc/269982},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Ryszard Jajte
TI - Convergence of orthogonal series of projections in Banach spaces
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 137
EP - 153
AB - 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 ∈ (A)).
LA - eng
KW - idempotent; mutually orthogonal projections; L₂-space; convergence almost everywhere; -space; idempotent operators
UR - http://eudml.org/doc/269982
ER -

## References

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