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<p>We establish the relationship between the S-spectrum of <i>fg</i> and the S-spectrum of α<i>f</i> + β<i>g,</i> where <i>f</i> and <i>g</i> are two idempotents on a two-sided quaternionic Banach algebra 𝒜 and α<i>,</i> β ∈ ℝ<i>.</i> We also study the invertibility of linear combinations of idempotents on 𝒜<i>.</i> To do this, we define the complexification 𝒜<sub><i>c</i></sub> of 𝒜<i>,</i> which is regarded as a real Banach algebra, and establish the relationship between the S-spectrum of <i>a</i> ∈ 𝒜 and the spectrum of <i>a</i> as an element of the complexification 𝒜<sub><i>c</i></sub><i>.</i></p>

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The S-Spectrum of Combinations of Idempotents on Two-Sided Quaternionic Banach Algebra

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Somayya Moulaharabbi,
  • Mohamed Barraa

摘要

We establish the relationship between the S-spectrum of fg and the S-spectrum of αf + βg, where f and g are two idempotents on a two-sided quaternionic Banach algebra 𝒜 and α, β ∈ ℝ. We also study the invertibility of linear combinations of idempotents on 𝒜. To do this, we define the complexification 𝒜c of 𝒜, which is regarded as a real Banach algebra, and establish the relationship between the S-spectrum of a ∈ 𝒜 and the spectrum of a as an element of the complexification 𝒜c.