Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Operator Inequalities Related to Angular Distances

  • Taba, Davood Afkhami (Department of Mathematics, Islamic Azad University) ;
  • Dehghan, Hossein (Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS))
  • Received : 2017.03.25
  • Accepted : 2017.08.04
  • Published : 2017.12.23
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Abstract

For any nonzero elements x, y in a normed space X, the angular and skew-angular distance is respectively defined by ${\alpha}[x,y]={\parallel}{\frac{x}{{\parallel}x{\parallel}}}-{\frac{y}{{\parallel}y{\parallel}}}{\parallel}$ and ${\beta}[x,y]={\parallel}{\frac{x}{{\parallel}y{\parallel}}}-{\frac{y}{{\parallel}x{\parallel}}}{\parallel}$. Also inequality ${\alpha}{\leq}{\beta}$ characterizes inner product spaces. Operator version of ${\alpha}$ has been studied by $ Pe{\check{c}}ari{\acute{c}}$, $ Raji{\acute{c}}$, and Saito, Tominaga, and Zou et al. In this paper, we study the operator version of ${\beta}$ by using Douglas' lemma. We also prove that the operator version of inequality ${\alpha}{\leq}{\beta}$ holds for commutating normal operators. Some examples are presented to show essentiality of these conditions.

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References

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