# Norm of sum of self-adjoint operators

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.

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