On some operator norm inequalities (Q1887616)

Let \(B(H)\) be the \(C^{*}\)-algebra of all bounded linear operators on a complex Hilbert space \(H\), and let \((I, \| \cdot\| _{I})\) denote a norm ideal of \(B(H)\). For \(A, B\in B(H)\), the author defines the two operators \(U_{I,A,B}(X)=AXB+BXA\) and \(V_{I,A,B}(X)=AXB-BXA\) on \(I\). Then the author obtains lower and upper bounds of \(U_{I,S,S^{-1}}\) and \(V_{I,S,S^{-1}}\) in the case that \(S\) is an invertible and self-adjoint operator. Using this result, the author obtains the following norm inequality: \[ \| SXS^{-1}-S^{-1}XS\| _{I} \leq (\| S\| \| S^{-1}\| -1) \| SXS^{-1}+S^{-1}XS\| _{I} \] for \(X\in I\) and \(S=S^{*}\). This is another type of the inequality shown in [\textit{F. Kittaneh}, Linear Algebra Appl. 208--209, 19--28 (1994; Zbl 0803.47019)].