Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and in measure for some f ∈ M(μ;X), then also in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying f = (T-I)h.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-9,
author = {Ryotaro Sato},
title = {Solvability of the functional equation f = (T-I)h for vector-valued functions},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {253-265},
zbl = {1072.47010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-9}
}
Ryotaro Sato. Solvability of the functional equation f = (T-I)h for vector-valued functions. Colloquium Mathematicae, Tome 100 (2004) pp. 253-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-2-9/