This is an announcement for the paper "Proper subspaces and compatibility" by Esteban Andruchow, Eduardo Chiumiento and Maria Eugenia Di Iorio y Lucero.
Abstract: Let $\mathcal{E}$ be a Banach space contained in a Hilbert space $\mathcal{L}$. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambicki\v{\i}, we say that a bounded operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with respect to the inner product of $\mathcal{L}$. By a proper subspace $\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $\mathcal{L}$, then $\mathcal{S}$ 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 to ensure that a proper subspace is compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$ which is also a proper subspace. We give a characterization of the compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several examples are provided that illustrate different situations between proper and compatible subspaces.
Archive classification: math.FA
Remarks: 18 pages
Submitted from: eduardo@mate.unlp.edu.ar
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1503.00596
or