گراف ناجابجایی عملگرهای خطی کران دار روی یک فضای هیلبرت

ﭼﮑﻴﺪﻩ:ﻓﺮﺽ ﮐﻨﻴﻢh ﻳﮏ ﻓﻀﺎﻱ ﻫﻴﻠﺒﺮﺕ ﻣﺨﺘﻠﻂ ﻭ (b(h ﺟﺒﺮ ﺷﺎﻣﻞ ﺗﻤﺎﻡ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﺧﻄﻲ ﮐﺮﺍﻥ ﺩﺍﺭ ﺭﻭﻱh ﺑﺎﺷﺪ. ﮔﺮﺍﻑ ﻧﺎ ﺟﺎ ﺑﻪ ﺟﺎﻳﻲ (b(h ﮐﻪ ﺁﻥ ﺭﺍ ﺑﺎ ﻧﻤﺎﺩ ((γ (b(h ﻧﻤﺎﻳﺶ ﻣﻲ ﺩﻫﻴﻢ، ﮔﺮﺍﻓﻲ ﺳﺎﺩﻩ ﺍﺳﺖ ﮐﻪ ﻣﺠﻤﻮﻋﻪ ﺭﺃﺱ ﻫﺎﻱ ﺁﻥ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﺧﻄﻲ ﮐﺮﺍﻥ ﺩﺍﺭ ﻏﻴﺮ ﺍﺳﮑﺎﻟﺮ ﺭﻭﻱh ﻣﻲ ﺑﺎﺷﺪ ﻭ ﺩﻭ ﺭﺍﺱ ﻣﺘﻤﺎﻳﺰa ﻭb ﺭﺍ ﺑﻪ ﻳﮑﺪﻳﮕﺮ ﻭﺻﻞ ﻣﻲ ﮐﻨﻴﻢ، ﺍﮔﺮ ﻭ ﻓﻘﻂ ﺍﮔﺮab ̸= ba . ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ، ﻧﺸﺎﻥ ﻣﻲ ﺩﻫﻴﻢ ﺑﺮﺍﻱ ﻫﺮ ﻓﻀﺎﻱ ﻫﻴﻠﺒﺮﺕ ﻣﺨﺘﻠﻂ، ((γ (b(h ﮔﺮﺍﻓﻲ ﻫﻤﺒﻨﺪ ﺍﺳﺖ. ﻫﻢ ﭼﻨﻴﻦ ﺛﺎﺑﺖ ﻣﻲ ﮐﻨﻴﻢ ﮔﺮﺍﻑ ﻫﺎﻱ ﻧﺎ ﺟﺎ ﺑﻪ ﺟﺎﻳﻲ ﻓﻀﺎﻱ ﺷﺎﻣﻞ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﺑﺎ ﺭﺗﺒﻪ ﻣﺘﻨﺎﻫﻲ ﺭﻭﻱh ، ﻓﻀﺎﻱ ﺷﺎﻣﻞ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﻓﺸﺮﺩﻩ ﺭﻭﻱh ، ﻓﻀﺎﻱ ﺷﺎﻣﻞ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﻭﺍﺭﻭﻥ ﻧﺎﭘﺬﻳﺮ ﺭﻭﻱh ﻭ ﻓﻀﺎﻱ ﺷﺎﻣﻞ ﻋﻤﻞ ﮔﺮﻫﺎﻱ ﻓﺮﺩﻫُﻠﻢ ﺭﻭﻱ h، ﮔﺮﺍﻑ ﻫﺎﻳﻲ ﻫﻤﺒﻨﺪ ﻣﻲ ﺑﺎﺷﻨﺪ.

The Noncommuting Graph of Bounded Linear Operators on a Hilbert Space

Let $EuScript{H}$ be a Complex Hilbert space and $EuScript{B(H)}$ be the algebra of allbounded linear operators on $EuScript{H}$. The noncommuting graph of $EuScript{B(H)}$, denoted by $mathnormal{Gamma}(EuScript{B(H)})$ is a graph whose vertices are nonscalar 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 nonseparable 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 noninvertible operators on $EuScript{H}$ and the set of all Fredholm operators on $EuScript{H}$ are connected graphs.