Let A and B be positive operators on and let P be a projection. Is the inequality $$\|A+PBP\|\le\|A+B\|$$ true? Here $\|.\|$ stands for the operator norm.
2026-04-08 05:38:50.1775626730
Is it true that $\|A+PBP\|\le\|A+B\|$ for every projection $P$ and positive operators $A,B$?
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Consider the case of $\mathbb C^2$ with the matrices $$ A = \pmatrix{1 & 0\cr 0 & 6\cr},\ B = \pmatrix{6 & 0\cr 0 & 1\cr}, \ P = \pmatrix{1/2 & 1/2\cr 1/2 & 1/2\cr}$$