# Sur les isométries partielles maximales essentielles

Studia Mathematica (1998)

• Volume: 128, Issue: 2, page 135-144
• ISSN: 0039-3223

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## Abstract

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We study the problem of approximation by the sets S + K(H), ${S}_{e}$, V + K(H) and ${V}_{e}$ where H is a separable complex Hilbert space, K(H) is the ideal of compact operators, $S=L\in B\left(H\right):L*L=I$ is the set of isometries, V = S ∪ S* is the set of maximal partial isometries, ${S}_{e}=L\in B\left(H\right):\pi \left(L*\right)\pi \left(L\right)=\pi \left(I\right)$ and ${V}_{e}={S}_{e}\cup {S}_{e}*$ where π : B(H) → B(H)/K(H) denotes the canonical projection. We also prove that all the relevant distances are attained. This implies that all these classes are closed and we remark that ${V}_{e}=V+K\left(H\right)$. We also show that S + K(H) is both closed and open in ${S}_{e}$. Finally, we prove that ${V}_{e}$, S + K(H) and ${S}_{e}$ coincide with their boundaries $\partial \left({V}_{e}\right)$, ∂(S + K(H)) and $\partial \left({S}_{e}\right)$ respectively.

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