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## Studia Mathematica

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## Proper subspaces and compatibility

### Tom 231 / 2015

Studia Mathematica 231 (2015), 195-218 MSC: Primary 46B20; Secondary 47A05, 47A30. DOI: 10.4064/sm8225-2-2016 Opublikowany online: 29 February 2016

#### Streszczenie

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.

#### Autorzy

• Esteban AndruchowInstituto 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
e-mail
• Eduardo ChiumientoDepartamento 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
e-mail
• María Eugenia Di Iorio y LuceroInstituto 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
e-mail

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