Let X be a two-sided Banach quaternionic space and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A, C, B, D: X \rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>D</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> be the right bounded linear operators satisfying operator equation set <Equation ID="Equ1"> <EquationSource Format="TEX">$\begin{aligned} A C D=D B D \;and\; D B A=A C A. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>A</mi> <mi>C</mi> <mi>D</mi> <mo>=</mo> <mi>D</mi> <mi>B</mi> <mi>D</mi> <mspace width="0.277778em" /> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mspace width="0.277778em" /> <mi>D</mi> <mi>B</mi> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mi>C</mi> <mi>A</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we generalize Jacobson’s Lemma and investigate the common properties of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(A C)^2-2 \operatorname {Re}(q) A C+|q|^2 I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mo>Re</mo> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> <mi>C</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>q</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(B D)^2-2 \operatorname {Re}(q) B D+|q|^2 I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mo>Re</mo> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> <mi>B</mi> <mi>D</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>q</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> where I stands for the identity operator on X and non-zero quaternion q. In particular, we show that <Equation ID="Equ2"> <EquationSource Format="TEX">$ \sigma _{\mathcal {}}^S(A C) \backslash \{0\}=\sigma _{\mathcal {}}^S(B D) \backslash \{0\}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>σ</mi> <mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mrow> <mi>S</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="true">\</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> <mo>=</mo> <msubsup> <mi>σ</mi> <mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mrow> <mi>S</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> </mrow> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma _{\mathcal {*}}^S(.)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>σ</mi> <mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mrow> <mi>S</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a distinguished part of the spherical spectrum.

Extensions of Jacobson’s Lemma and Cline’s formula in the Quaternionic Setting

期刊论文
发表日期：

2026年4月11日

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Aziz Blali,
Abdellah El Allaoui,
Abdelkhalek El Amrani

摘要

Let X be a two-sided Banach quaternionic space and $A, C, B, D: X \rightarrow X$ $A, C, B, D : X \to X$ be the right bounded linear operators satisfying operator equation set $\begin{aligned} A C D=D B D \;and\; D B A=A C A. \end{aligned}$ $\begin{matrix} A C D = D B D a n d D B A = A C A . \end{matrix}$ In this paper, we generalize Jacobson’s Lemma and investigate the common properties of $(A C)^2-2 \operatorname {Re}(q) A C+|q|^2 I$ ${(A C)}^{2} - 2 Re (q) A C + {| q |}^{2} I$ and $(B D)^2-2 \operatorname {Re}(q) B D+|q|^2 I$ ${(B D)}^{2} - 2 Re (q) B D + {| q |}^{2} I$ where I stands for the identity operator on X and non-zero quaternion q. In particular, we show that $ \sigma _{\mathcal {*}}^S(A C) \backslash \{0\}=\sigma _{\mathcal {*}}^S(B D) \backslash \{0\}, $ $σ_{*}^{S} (A C) \ {0} = σ_{*}^{S} (B D) \ {0},$ where $\sigma _{\mathcal {*}}^S(.)$ $σ_{*}^{S} (.)$ is a distinguished part of the spherical spectrum.