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Best approximations in a class of Lorentz ideals. (English) Zbl 1502.41009

Summary: We consider the family of Lorentz ideals \(\mathcal{C}_p^+\), \(1 \le p < \infty\). Let \(\mathcal{C}_p^{+(0)}\) be the \(\Vert \cdot \Vert_p^+\)-closure of the collection of finite-rank operators in \(\mathcal{C}_p^+\). It is well known that \(\mathcal{C}_p^{+(0)} \ne\mathcal{C}_p^+\). We show that \(\mathcal{C}_p^{+(0)}\) is proximinal in \(\mathcal{C}_p^+\). We further show that a classic approximation for Hankel operators [S. Axler et al., Ann. Math. (2) 109, 601–612 (1979; Zbl 0399.47024)] does not generalize to this new context.

MSC:

41A50 Best approximation, Chebyshev systems
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators

Citations:

Zbl 0399.47024
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Full Text: DOI

References:

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