Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space $E$. On $\mathcal{L}(E)^2$, we have two equivalent norms: \begin{eqnarray*} N_1(A,B) &=&\sup\left\{\|Ax\|^2+\|Bx\|^2,\;x\in E,\;\|x\|=1\;\right\}, \end{eqnarray*} and $$N_2(A,B)=\|A\|^2+\|B\|^2.$$
Clearly, $N_1(A,B)\leq N_2(A,B)$.
In general $N_1\neq N_2$. I want to find special cases for equality. I claim that: if $AB=BA$ and $A$ et $B$ are normal operators on $E$, then $$N_1(A,B)= N_2(A,B).$$ Is my claim correct?
I doubt. Consider $E=\mathbb{C}^2$, $A=\rm{diag}(1,0)$ and $B=\rm{diag}(0,1)$. Then $A$ and $B$ are selfadjoint, hence normal, and $AB=BA=0$. Now $$ N_1(A,B)=\sup\{\|A{x\choose y}\|^2 +\|B{x\choose y}\|^2: ~ {x\choose y} \in E, ~ |x|^2+|y|^2=1\}=1, $$ but $N_2(A,B)=\|A\|^2+\|B\|^2=2$.