This is a cross-post from [math.stackexchange.com](https://math.stackexchange.com/questions/2696441/symmetric-c-property)


What can be said about elements $a$ of a C\*-algebra which fulfil the 'symmetric C\* property'
$$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$


I'd guess that this is not a general property, but I don't have a good idea which elements (except for self-adjoint and normal ones, of course) should satisfy this. 



Symmetric C* property
This is hardly a complete answer.


As you say, all normal operators $a \in B(\mathcal H)$ satisfy this equality. Moreover, so does any element $c = a\oplus b \in B(\mathcal H\oplus \mathcal H)$ where $a$ is normal and $\|b\| \leq \|a\|$. This is because
$$ \|c^\*c + cc^\*\| = \max\{\|a^\*a + aa^\*\|, \|b^\*b + bb^\*\|\} = 2\|a\| = 2\|c\|. $$


On the other hand, this is certainly not a general property. For instance, if $a = \left[\begin{matrix} 0 & 1\\ 0&0\end{matrix}\right]$ then $$\|a^\*a + a^\*a\| = \left\|\left[\begin{matrix} 1 & 0\\ 0&1\end{matrix}\right]\right\| = 1 \quad \textrm{but} \quad 2\|a\|^2 = 2.$$


And on the third hand, $a = \left[\begin{matrix} 0 & 1 & 0 \\ 0& 0& 1\\ 0&0&0\end{matrix}\right]$ satisfies this equality. This makes me doubt that there will be any nice characterization.


