Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.

For $A= (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$ (not necessary to be commuting). Why $$\lim_{n\to+\infty}\bigg(\bigg\|\sum_{f\in F(n,d)} A_{f}^* A_{f}\bigg\|^{\frac{1}{2n}} \bigg)\;\text{exists}?,$$ where $F(n,d)$ denotes the set of all functions from $\{1,\cdots,n\}$ into $\{1,\cdots,d\}$ and $A_f:=A_{f(1)}\cdots A_{f(n)}$, for $f\in F(n,d)$.

I try to apply Fekete's lemma: For every subadditive sequence $(a_n)_{n\in \mathbb{N}^*}$, the limit $\displaystyle \lim _{n\to \infty }\frac{a_n}{n}$ exists and is equal to $\displaystyle \inf_{n\in \mathbb{N}^*}\frac{a_n}{n}$. (The limit may be ${\displaystyle -\infty }$).

But I'm facing difficulties in the choice of the sequence $(a_n)_{n\in \mathbb{N}^*}$.

Thank you for your help.