Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties

This article presents a characterization of non-singular measurable transformation, denoted as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T , mapping from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> </m:math> Ω to itself, along with bicomplex-valued <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">BC</m:mi> </m:math> {BC} -measurable function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> u defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ω</m:mi> </m:math> Ω , which induces a weighted composition operator. The study then proceeds to fully identify their <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">D</m:mi> </m:math> {D} -compactness and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">D</m:mi> </m:math> {D} -closedness within the range of bicomplex Lorentz spaces denoted as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="double-struck">BC</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="fraktur">M</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="italic">ϑ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {L}p,q^{{BC}}(Ω ,{M},{ϑ }) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="fraktur">M</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="italic">ϑ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Ω ,{M},{ϑ }) represents a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>σ</m:mi> </m:math> σ -finite complete <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">BC</m:mi> </m:math> {BC} -measure space, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="italic">ϑ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="italic">ϑ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="italic">ϑ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {ϑ }={{ϑ }}₁{e}₁+{{ϑ }}₂{e}₂ is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">BC</m:mi> </m:math> {BC} -measure, and the parameters satisfy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> </m:math> 1< p≤ ∞ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> </m:math> 1≤ q≤ ∞ .