Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8225-2-2016,
author = {Esteban Andruchow and Eduardo Chiumiento and Mar\'\i a Eugenia Di Iorio y Lucero},
title = {Proper subspaces and compatibility},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {195-218},
zbl = {06575013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8225-2-2016}
}
Esteban Andruchow; Eduardo Chiumiento; María Eugenia Di Iorio y Lucero. Proper subspaces and compatibility. Studia Mathematica, Tome 231 (2015) pp. 195-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8225-2-2016/