# Prove an equality related to a tuple of operators

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?

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 $A_1A_1^*+A_2A_2^*=1$.