Proper subspaces and compatibility
Volume 231 / 2015
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 Zambickiĭ, 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 $\mathcal {L}$. A proper operator which is self-adjoint with respect to the inner product of $\mathcal {L}$ is called symmetrizable. By a proper subspace $\mathcal {S}$ we mean a closed subspace of $\mathcal {E}$ which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto $\mathcal {S}$, 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 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 $\mathcal {S}$ 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.
