Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

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, $S = {L ∈ B(H) : L*L = I}$ is the set of isometries, V = S ∪ S* is the set of maximal partial isometries, $S_e = {L ∈ B(H): π(L*)π( L) = π(I)}$ and $V_e = S_e ∪ S_e*$ 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 $V_e = V + K(H)$. 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 $∂(V_e)$, ∂(S + K(H)) and $∂(S_e)$ respectively.