Proper subspaces and compatibility

Abstract

Let E be a Banach space contained in a Hilbert space L. Assume thatthe inclusion is continuous with dense range. Following the terminology of Gohberg andZambicki, we say that a bounded operator on E is a proper operator if it admits anadjoint with respect to the inner product of L. A proper operator which is self-adjointwith respect to the inner product of L is called symmetrizable. By a proper subspace Swe 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 ofsubspaces called compatible subspaces. We nd equivalent conditions to describe propersubspaces. Then we prove a necessary and sucient condition for a proper subspace tobe compatible. The existence of non-compatible proper subspaces is related to spectralproperties of symmetrizable operators. Each proper subspace S has a supplement T whichis also a proper subspace.We give a characterization of the compatibility of both subspacesS and T . Several examples are provided that illustrate dierent situations between properand compatible subspaces