Berens, H.; Finzel, M. A problem in linear matrix approximation. (English) Zbl 0838.47015 Math. Nachr. 175, 33-46 (1995). 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. Cited in 2 Documents 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. Keywords:commutator subspace; spectral norm; Schatten \(p\)-norm; approximation theoretical point of view; elements of best approximation PDFBibTeX XMLCite \textit{H. Berens} and \textit{M. Finzel}, Math. Nachr. 175, 33--46 (1995; Zbl 0838.47015) Full Text: DOI References: [1] Anderson, Proc. Amer. Math. Soc. 38 pp 135– (1973) [2] and , A Continuous Selection of the Metric Projection in Matrix Spaces, in Proceedings on ”Numerical Methods of Approximation Theory”, Vol. 8 1986, , and eds., ISNM 81 (1987), 21–29 [3] Berens, addendum, Numer. Math. 57 pp 663– (1990) [4] Berens, ISNM 40 pp 119– (1977) [5] Halmos, Amer. J. Math. 76 pp 191– (1954) [6] and , Topics in Matrix Analysis, Cambridge University Press, Cambridge 1991 · Zbl 0729.15001 · doi:10.1017/CBO9780511840371 [7] Kadison, Ann. Math. 54 pp 325– (1951) [8] Maher,, Commutator Approximants, Proc. Amer. Math. Soc. 115 pp 995– (1992) [9] Topics in Approximation Theory, Lecture Notes in Math. Vol. 187, Springer Verlag, Berlin 1970 [10] Watson, Linear Algebra Appl. 170 pp 33– (1992) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.