Let $a$, $b$ be positive numbers. $A$ is a positive self-adjoint operator in a Hilbert space, such that $aI \leqslant A \leqslant bI$. How to prove $b^{-1}I \leqslant A^{-1} \leqslant a^{-1}I$?
2026-03-25 14:28:50.1774448930
Self-adoint positive operator $A$
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Hint: If $aI\leq A\leq bI$, then $\sigma(A)\subset [a,b]$. What does this say about $\sigma(A^{-1})$?