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Tytuł artykułu

Proper subspaces and compatibility

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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 Zambickiĭ, 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. 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 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 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 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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  • Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez 1150, 1613 Los Polvorines, Argentina
  • Instituto Argentino de Matemática, 'Alberto P. Calderón', CONICET, Saavedra 15, 3er. piso, 1083 Buenos Aires, Argentina
  • Departamento de Matemática, FCE-UNLP, Calles 50 y 115, 1900 La Plata, Argentina
  • Instituto Argentino de Matemática, 'Alberto P. Calderón', CONICET, Saavedra 15, 3er. piso, 1083 Buenos Aires, Argentina
  • Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez 1150, 1613 Los Polvorines, Argentina
  • Instituto Argentino de Matemática, 'Alberto P. Calderón', CONICET, Saavedra 15, 3er. piso, 1083 Buenos Aires, Argentina
Bibliografia
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Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-sm8225-2-2016
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