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• W2132317719 abstract "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 Zambickivi, 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 $mathcal{L}$. By a proper subspace $mathcal{S}$ we mean a closed subspace of $mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $mathcal{L}$, 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 to ensure that a proper subspace is compatible. 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." @default.
• W2132317719 created "2016-06-24" @default.
• W2132317719 creator A5054718262 @default.
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• W2132317719 date "2015-03-02" @default.
• W2132317719 modified "2023-10-06" @default.
• W2132317719 title "Proper subspaces and compatibility" @default.
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• W2132317719 hasPublicationYear "2015" @default.
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