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A problem in linear matrix approximation. (English) Zbl 0838.47015

Summary: Let \(A\) be a normal operator in \({\mathcal B} (H)\), \(H\) a complex Hilbert space, and let \({\mathcal R}_A= \{AX- XA: X\in {\mathcal B}(H)\}\) be the commutator subspace of \({\mathcal B} (H)\) associated with \(A\). If \(B\) in \({\mathcal B}(H)\) commutes with \(A\), then \(B\) is orthogonal to \({\mathcal R}_A\) with respect to the spectral norm; i.e., the null operator is an element of best approximation of \(B\) in \({\mathcal R}_A\). This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten \(p\)-norm recently. We take a look at their result from a more approximation theoretical point of view in the finite-dimensional setting; in particular, we characterize all elements of best approximation of \(B\) in \({\mathcal R}_A\) and prove that the metric projection of \(H\) onto \({\mathcal R}_A\) is continuous.

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
41A50 Best approximation, Chebyshev systems
41A35 Approximation by operators (in particular, by integral operators)
47B47 Commutators, derivations, elementary operators, etc.
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References:

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