Proper subspaces and compatibility

ANDRUCHOW ESTEBAN; CHIUMIENTO EDUARDO; DI IORIO Y LUCERO, EUGENIA

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2015 vol. 231 p. 195 - 218

0039-3223

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 Zambickii, we say that a bounded operator on E is a proper operatorif it admits an adjoint with respect to the inner product of L. A proper operator which is self-adjoint with respect to the inner product of L is called symmetrizable. By a proper subspace S we mean a closed subspace of E whichis the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto S, 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. The existence of non compatible proper subspaces is related to spectral properties of symmetrizable operators. 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.