article · Proceedings of the American Mathematical Society · 1992

Commutator Approximants

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Abstract

This paper deals with minimizing $||B - (AX - XA)|{|_p}$, where $A$ and $B$ are fixed, $B \in {\mathcal {C}_p}$, and $X$ varies such that $AX - XA \in {\mathcal {C}_p}$. (Here, ${\mathcal {C}_p}$ denotes the von Neumann-Schatten class and ${\left \| \right \|_p}$ denotes its norm.) The main result (Theorem 3.2) says that if $A$ is normal and $AB = BA$ then $||B - (AX - XA)|{|_p},1 \leq p < \infty$, is minimized if and for $1 < p < \infty$ only if, $AX - XA = 0$; and that the map $X \to ||B - (AX - XA)||_p^p,1 < p < \infty$, has a critical point at $X = V$ if and only if $AV - VA = 0$.

Cite this paper

APA 7

Maher, P. (1992). Commutator Approximants. Proceedings of the American Mathematical Society, 115(4), 995. https://doi.org/10.2307/2159345

MLA 9

Maher, Philip. “Commutator Approximants.” Proceedings of the American Mathematical Society, vol. 115, no. 4, 1992, p. 995, https://doi.org/10.2307/2159345.

Chicago

Maher, Philip. 1992. “Commutator Approximants.” Proceedings of the American Mathematical Society 115 (4): 995. https://doi.org/10.2307/2159345.

Harvard

Maher, P. (1992) ‘Commutator Approximants’, Proceedings of the American Mathematical Society, 115(4), p. 995. Available at: https://doi.org/10.2307/2159345.

BibTeX

@article{maher1992commutator,
  title = {{Commutator Approximants}},
  author = {Maher, Philip},
  journal = {Proceedings of the American Mathematical Society},
  volume = {115},
  number = {4},
  pages = {995},
  year = {1992},
  doi = {10.2307/2159345}
}

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