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# Journal of Operator Theory

Volume 48, Issue 1, Summer 2002  pp. 95-103.

Norm inequalities for sums of positive operators

Summary:  We use certain norm inequalities for $2\times 2$ operator matrices to establish norm inequalities for sums of positive operators. Among other inequalities, it is shown that if $A$ and $B$ are positive operators on a Hilbert space, then $$\| A+B\| \leq {1\over 2} \Big( \|A\|+\|B\|+\sqrt{(\|A\|-\|B\|)^2 +4 \|A^{1/2}B^{1/2}\|^2}\Big).$$ This inequality, which is sharper than the triangle inequality, improves upon some earlier related inequalities. Applications of these inequalities are also considered.