The Noncommuting Graph of Bounded Linear Operators on a Hilbert Space

Author

Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

Abstract

Let $EuScript{H}$ be a Complex Hilbert space and $EuScript{B(H)}$ be the algebra of all

bounded linear operators on $EuScript{H}$. The non-commuting graph of $EuScript{B(H)}$, denoted by $mathnormal{Gamma}(EuScript{B(H)})$ is a graph whose vertices are non-scalar bounded operators and two distinct vertices $A$ and $B$ are adjacent if and only if $AB neq BA$. In this paper, we prove the connectivity of $mathnormal{Gamma}(EuScript{B(H)})$ for separable and non-separable complex Hilbert spaces. Also we show that the noncommuting graphs of the set of all finite rank operators on $EuScript{H}$, the set of all compact operators on $EuScript{H}$, the set of all non-invertible operators on $EuScript{H}$ and the set of all Fredholm operators on $EuScript{H}$ are connected graphs.

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References

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Journal of Advanced Mathematical Modeling
Volume 12, Issue 2
June 2022
Pages 168-174

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History

  • Receive Date: 11 December 2021
  • Revise Date: 17 April 2022
  • Accept Date: 24 April 2022

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