In this paper, we establish several inequalities concerning to the Moore–Penrose geometric quasi-mean <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1260_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="TEX">$A\textcircled {p}_{\nu }B=||BA^\dag |^\nu A|^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <menclose notation="circle"> <mi mathvariant="normal">p</mi> </menclose> <mi>ν</mi> </msub> <mi>B</mi> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi>B</mi> </mrow> <msup> <mi>A</mi> <mo>†</mo> </msup> <msup> <mrow> <msup> <mo stretchy="false">|</mo> <mi>ν</mi> </msup> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> for bounded conditional operator A and B in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1260_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2(\Sigma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where A has closed range and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1260_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, we discuss some measure theoretic characterizations for conditional operators in some operator classes.

Generalized inverse and geometric mean of Lambert conditional operators

期刊论文
发表日期：

2025年7月2日

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M. R. Jabbarzadeh,
M. Sohrabi

摘要

In this paper, we establish several inequalities concerning to the Moore–Penrose geometric quasi-mean $A\textcircled {p}_{\nu }B=||BA^\dag |^\nu A|^2$ $A p_{ν} B = | | B A^{†} {|^{ν} A |}^{2}$ for bounded conditional operator A and B in $L^2(\Sigma )$ $L^{2} (Σ)$ , where A has closed range and $\nu \ge 0$ $ν \geq 0$ . In addition, we discuss some measure theoretic characterizations for conditional operators in some operator classes.