On convexity and compactness of operator ``intervals'' on Hilbert space

Abstract

We consider a von Neumann algebra $M$ acting on a Hilbert space $H$. For a positive operator $X$ in $M$ we define the operator ``intervals'' $I_X=\{Y=Y^*\in M: \; -X \leq Y \leq X \}$ and $L_X=\{Y \in M: \; |Y| \leq X \}$, where $|Y|=\sqrt{Y^*Y}$. The properties of this operator ``intervals'' are investigated. We prove that a von Neumann algebra $M$ is Abelian if and only if $L_X$ is convex for all $X$ in $M$. We then show for $M=B(H)$, the algebra of all linear bounded operators on $H$, that the operator ``interval'' $I_X$ is compact if and only if an operator $X$ is compact.