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Let $L$ and $K$ be such self-adjoint operators on a Hilbert space, such that we have $LK=0$. Show that for the operator norm, we have the equality

$$\left\|L+K\right\|=\max\left\{\left\|L\right\|, \left\|K\right\|\right\}.$$

I was able to prove that $\left\|L\right\|\leq \left\|L+K\right\|$ and $\left\|K\right\| \leq \left\|L+K\right\|,$ but I cannot prove the other inequality.

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Hint: show that $$ H=(\ker K)^\perp\oplus(\ker L)^\perp\oplus(\ker K\cap \ker L) $$ and with respect to this decomposition, $$ K=\begin{pmatrix}K\vert_{\overline{\operatorname{im}K}}\\&0\\&&0\end{pmatrix}, L=\begin{pmatrix}0\\&L\vert_{\overline{\operatorname{im}L}}\\&&0\end{pmatrix}. $$

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