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W2072636811 abstract "In this paper, we present an extension of Bouldin’s result (1970) concerning the numerical range <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W left-parenthesis upper A upper B right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W(AB)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the product of two operators <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> that are commuting and for which one of the set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W left-parenthesis upper B right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W(B)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consists of positive numbers. We also prove 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> or <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> is a subnormal operator on a separable Hilbert space, then <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper W left-parenthesis upper M Subscript 2 comma upper A comma upper B Baseline right-parenthesis With bar equals ModifyingAbove c o left-bracket upper W left-parenthesis upper A right-parenthesis upper W left-parenthesis upper B right-parenthesis right-bracket With bar comma> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo accent=false>¯</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> <mml:mo accent=false>¯</mml:mo> </mml:mover> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} overline {W(M_{2,A,B})}=overline {coleft [ W(A)W(B)right ] }, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Subscript 2 comma upper A comma upper B> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>M_{2,A,B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the operator bimultiplication and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>co</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the convex hull. <sc>Résumé</sc>. Dans ce travail, nous améliorons un résultat de Bouldin (1970) concernant la localisation de <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W left-parenthesis upper A upper B right-parenthesis comma> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W(AB),</mml:annotation> </mml:semantics> </mml:math> </inline-formula> le domaine numérique du produit de deux opérateurs <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> et <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> sur un espace de Hilbert lorsque <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> et <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> commutent et <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> est constitué de réels strictement positifs. Dans le cas où <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> ou <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> est un opérateur sous normal sur un espace de Hilbert séparable, nous montrons que <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ModifyingAbove upper W left-parenthesis upper M Subscript 2 comma upper A comma upper B Baseline right-parenthesis With bar equals ModifyingAbove c o left-bracket upper W left-parenthesis upper A right-parenthesis upper W left-parenthesis upper B right-parenthesis right-bracket With bar comma> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo accent=false>¯</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mover> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> <mml:mo accent=false>¯</mml:mo> </mml:mover> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} overline {W(M_{2,A,B})}=overline {coleft [ W(A)W(B)right ] }, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> où <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M Subscript 2 comma upper A comma upper B> <mml:semantics> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>M_{2,A,B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> est l’opérateur produit ou bimultiplication et <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c o> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>co</mml:annotation> </mml:semantics> </mml:math> </inline-formula> est l’enveloppe convexe." @default.
W2072636811 created "2016-06-24" @default.
W2072636811 creator A5013900107 @default.
W2072636811 date "2004-03-03" @default.
W2072636811 modified "2023-10-16" @default.
W2072636811 title "Domaine numérique du produit et de la bimultiplication 𝑀_{2,𝐴,𝐵}" @default.
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