Proper subspaces and compatibility
Volume 231 / 2015
Abstract
Let 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.