On Narrow Operators from $$L_p$$ into Operator Ideals

Jinghao Huang, Marat Pliev, Fedor Sukochev

Type: Article

Publication Date: 2022-08-30

Citations: 5

DOI: https://doi.org/10.1007/s00009-022-02118-y

Abstract

Abstract It is well known that every $$l_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -strictly singular operator from $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> , $$1<p<\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> to any Banach space X with an unconditional basis is narrow. In this article, we extend this result to the setting of Banach spaces without an unconditional basis. We show that if $$1 \le p,r <\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , then every $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -strictly singular operator T from $$L_p $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> into the Schatten–von Neumann r -class $$C_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:math> is narrow. This is a noncommutative complement to results in Mykhaylyuk et al. (in Israel J Math 203:81–108, 2014).

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Mediterranean Journal of Mathematics - View - PDF

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+	Harmonic analysis in (UMD)-spaces: Applications to the theory of bases	1995	Fedor Sukochev S. Ferleger
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+	On sign embeddings and narrow operators on 𝐿₂	2017	Beata Randrianantoanina