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Let $B(H) $ be the Banach algebra of all bounded linear operators acting on an infinite dimensional complex Hilbert space $ H $. Let $ A, B\in B(H) $, such that

$$ r(A)= \Vert A \Vert,$$ $$ r(B)= \Vert B \Vert .$$ Where $r(A)$ is the specral radius of $A$.

Do we have $$ r(A+B)= \Vert A +B\Vert?$$

A. Bag
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  • Can you give an example of $H$ and $C\in B(H)$ such that $r(C) <|C|$? Then we should only find an $A$ such that both $r(A) = |A|$ and $r(A+C) =|A+C|$.. – Berci Mar 13 '18 at 11:17

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Let $$A=\begin{pmatrix}10&0&0\\0&0&1\\0&0&0\end{pmatrix},\qquad B=\begin{pmatrix}-10&0&0\\0&0&0\\0&0&0\end{pmatrix}.$$ You have $\|A\|=\|B\|=10=r(A)=r(B)$. However $$A+B=\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix},$$ which has spectral radius $0$ but certainly not norm $0$. This example lives in $\Bbb C^3$, but just extend by $0$ on the orthogonal complement to get it to work on $B(H)$.

s.harp
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