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article · Proceedings of the American Mathematical Society · 1992

Commutator approximants

Philip Maher

Cited by 18Open access

Abstract

This paper deals with minimizing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">||B - (AX - XA)|{|_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are fixed, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B element-of script upper C Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">B \in {\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> varies such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper X minus upper X upper A element-of script upper C Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">AX - XA \in {\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {C}_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the von Neumann-Schatten class and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar double-vertical-bar Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\left \| \right \|_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes its norm.) The main result (Theorem 3.2) says that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is normal and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper B equals upper B upper A"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mi>B</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">AB = BA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline comma 1 less-than-or-equal-to p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">||B - (AX - XA)|{|_p},1 \leq p &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is minimized if and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 &gt; p &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only if, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper X minus upper X upper A equals 0"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">AX - XA = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and that the map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Superscript p Baseline comma 1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>B</mml:mi> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X \to ||B - (AX - XA)||_p^p,1 &gt; p &gt; \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, has a critical point at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals up

Cite this paper

APA 7

Maher, P. (1992). Commutator approximants. Proceedings of the American Mathematical Society, 115(4), 995–1000. https://doi.org/10.1090/s0002-9939-1992-1086335-6

MLA 9

Maher, Philip. “Commutator approximants.” Proceedings of the American Mathematical Society, vol. 115, no. 4, 1992, pp. 995–1000, https://doi.org/10.1090/s0002-9939-1992-1086335-6.

Chicago

Maher, Philip. 1992. “Commutator approximants.” Proceedings of the American Mathematical Society 115 (4): 995–1000. https://doi.org/10.1090/s0002-9939-1992-1086335-6.

Harvard

Maher, P. (1992) ‘Commutator approximants’, Proceedings of the American Mathematical Society, 115(4), pp. 995–1000. Available at: https://doi.org/10.1090/s0002-9939-1992-1086335-6.

BibTeX
@article{maher1992commutator,
  title = {{Commutator approximants}},
  author = {Maher, Philip},
  journal = {Proceedings of the American Mathematical Society},
  volume = {115},
  number = {4},
  pages = {995--1000},
  year = {1992},
  doi = {10.1090/s0002-9939-1992-1086335-6}
}

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