Let $E$ be an infinite-dimensional complex Hilbert space.
The spectral radius of a commuting multivariable operator $A = (A_1,\cdots,A_n)\in\mathcal{L}(E)^n$ is given by $$r_a(A_1,\cdots,A_n)=\displaystyle\lim_{m\to \infty}\left\|\displaystyle\sum_{|\alpha|=m}\frac{m!}{\alpha!}{A^*}^{\alpha}A^{\alpha}\right\|^{\frac{1}{2m}},$$ where $m\in\mathbb{N}^*,\;$ $\alpha = (\alpha_1, \alpha_2,...,\alpha_n) \in \mathbb{Z}_+^n;\;\alpha!: =\alpha_1!\cdots\alpha_n!,\;|\alpha|:=\displaystyle\sum_{j=1}^n|\alpha_j|$; $A^*=(A_1^*,\cdots,A_n^*)$ and $A^\alpha:=A_1^{\alpha_1} A_2^{\alpha_2}\cdots A_n^{\alpha_n}$.
I claim that if $A_iA_j=A_jA_i$ for all $i,j$, then in general $$r_a(A_1,\cdots,A_n)\neq r_a(A_1^*,\cdots,A_n^*).$$ I hope to find an explicit example which show that the claim is true.
Note also that we have \begin{align*} r_a(A_1,\cdots,A_n) &=\sup\{\|\lambda\|_2,\;\;\lambda \in \sigma_{ap}(A)\}, \end{align*} where $$\sigma_{ap}(A)=\bigg\{\lambda\in \mathbb{C}^n: \;\exists\;(x_k)_k\subset E;\,\,\|x_k\|=1\;\;\hbox{such that}\;\;\\\lim_{k\longrightarrow \infty}\sum_{1\leq j\leq n}\|(A_j-\lambda_j)x_k\|=0\bigg\}.$$
If $n=2$ and $A_1A_2=A_2A_1$, then $$r(A_1,A_2)=\displaystyle\lim_{m\longrightarrow +\infty}\left\{\sup_{\|x\|=1}\left[\displaystyle\sum_{p=0}^m{m\choose p}\|A_1^{m-p}A_2^{p}x\|^2\right]^{\frac{1}{2m}}\right\}.$$
Take $A=(A_1,A_2)$ such that $A_1=A_2$ and $A_1$ is neither self-adjoint nor normal. In this case $$r(A_1,A_2)\neq r(A_1^*,A_2^*).$$