<p>In this paper, we consider the unbounded case of substitution conditional operator <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2428_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\mapsto wE(u(f\circ \varphi ))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>↦</mo> <mi>w</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2428_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2(\mathcal {A})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and then we discuss some measure theoretic characterizations such as well-definiteness, closedness, dense definiteness and boundedness of <i>T</i>.</p>

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Unbounded Substitution Conditional Type Operators on \(L^2\)-Spaces

  • M. R. Jabbarzadeh

摘要

In this paper, we consider the unbounded case of substitution conditional operator \(f\mapsto wE(u(f\circ \varphi ))\) f w E ( u ( f φ ) ) on \(L^2(\mathcal {A})\) L 2 ( A ) and then we discuss some measure theoretic characterizations such as well-definiteness, closedness, dense definiteness and boundedness of T.