Abstract
For Hilbert space operatorsH,K,XwithH,K⩾0 the norm inequality |||H1/2XK1/2|||⩽12|||HX+XK||| is known, where |||·||| is an arbitrary unitarily invariant norm. A refinement of this arithmetic–geometric mean inequality is studied. Similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean.
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