Proper subspaces and compatibility

Esteban Andruchow ¹   ; Eduardo Chiumiento ²   ; María Eugenia Di Iorio y Lucero ¹

¹ Instituto de Ciencias Universidad Nacional de Gral. Sarmiento J.M. Gutierrez 1150 1613 Los Polvorines, Argentina and Instituto Argentino de Matemática ‘Alberto P. Calderón’ CONICET Saavedra 15, 3er. piso 1083 Buenos Aires, Argentina
² Departamento de Matemática, FCE-UNLP Calles 50 y 115 1900 La Plata, Argentina and Instituto Argentino de Matemática ‘Alberto P. Calderón’ CONICET Saavedra 15, 3er. piso 1083 Buenos Aires, Argentina

Studia Mathematica, Tome 231 (2015) no. 3, pp. 195-218

Résumé

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.

Détail

DOI : 10.4064/sm8225-2-2016

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