Abstract
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 Zambickiyo, 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. By a proper subspace S we mean a closed subspace of 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 L, 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 to ensure that a proper subspace is compatible. 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.
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