Let $E$ be a complex Hilbert space.


By applying Cauchy-Schwarz and elementary calculations, we prove that for all $(A\_1,...,A\_n) \in \mathcal{L}(E)^n$ we have
$$\sup\_{(\lambda\_1,...,\lambda\_n)\in B\_n}\left\|\sum\_{k=1}^n\lambda\_kA\_k\right\| \leq\left\|\sum\_{k=1}^nA\_kA\_k^\*\right\|^{1/2},$$
with $B\_n$ is the open unit ball of $\mathbb{C}^n$.



> 
> Is the following equality
>  $$\sup\_{(\lambda\_1,...,\lambda\_n)\in B\_n}\left\|\sum\_{k=1}^n\lambda\_kA\_k\right\| =\left\|\sum\_{k=1}^nA\_kA\_k^\*\right\|^{1/2},$$
>  always hold? If not, do expect that it holds if the operators $A\_k$ are **commuting**?
> 
> 
> 



Prove an equality related to a tuple of operators
If this were true, then we would have $\|\sum\_{k=1}^n A\_k^\*A\_k\|=\|\sum\_{k=1}^n A\_kA\_k^\*\|$. But there are examples when this doesn't hold. E.g., $A\_1$ and $A\_2$ isometries such that $A\_1A\_1^\*+A\_2A\_2^\*=1$.


