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- W4321456588 abstract "Abstract This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by $Smathcal{D}(L^{1})$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:math> ) on $L^{1}(X)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> when X is a σ -finite measure space. We answer Mirsky’s question and characterize the majorization by means of semidoubly stochastic maps on $L^{1}(X)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> . We prove some results on semidoubly stochastic operators such as a strong relation of semidoubly stochastic operators and integral stochastic operators and relatively weak compactness of $S_{f}={Sf : Sin Smathcal{D}(L^{1})}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mi>S</mml:mi> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:math> for a fixed element $fin L^{1}(X)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:math> by proving the equiintegrability of $S_{f}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> . We present a full characterization of majorization on a σ -finite measure space X ." @default.
- W4321456588 created "2023-02-22" @default.
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- W4321456588 date "2023-02-21" @default.
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- W4321456588 title "Majorization and semidoubly stochastic operators on $L^{1}(X)$" @default.
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