Let $H$ and $K$ be two Hilbert spaces equipped with orthonormal bases $\{h_{i,j}\}_{i,j\in\mathbb{Z}}$ and $\{k_{i,j}\}_{i=0,1,2,\ldots,\ j\in\mathbb{Z}}$ respectively and $\mu$ a positive real number smaller than $1$. Suppose $n$ and $\gamma$ are two normal operators on $H$ and $K$ respectively defined by $$n\ h_{i,j}=\mu^{i}\ h_{i,\ j+1}\quad \text{and} \quad\gamma\ k_{i,j}=\mu^{i}\ k_{i,\ j-1}$$Given the spectrum of the normal operator $n$ as $\text{sp}(n)=\{\lambda\in\mathbb{C}\ \colon|\lambda|\in\mu^{\mathbb{Z}}\}\cup\{0\} $ where $\mu^{\mathbb{Z}}=\{\mu^{i}\ |\ i\in\mathbb{Z}\}.$ Let $\Lambda=\text{sp}(n)\times \text{sp}(n)$, $f$ and $g$ be functions on $\Lambda$ such that $f(\lambda_1,\lambda_2)=|\lambda_1|^{2}$ and $g(\lambda_1,\lambda_2)=|\lambda_2|^{2}.$ For $R=\mu^{-2l}(l\in\mathbb{Z}),$ Set $\Lambda_{R}=\{\lambda\in\Lambda\ \colon\ f(\lambda)\leq R \text{ and } g(\lambda)\leq R\}.$ Let $d\rho(\lambda)=\langle\psi|dE(\lambda_1)\otimes dE(\lambda_2)|\psi\rangle$ where $dE(\cdot)$ is the spectral measure corresponding to the normal operator $n$ and $\psi\in H\otimes H.$
I need to show that $$\int_{\Lambda_{R}}f(\lambda)g(\lambda)d\rho(\lambda)=\mu^{-4l}\|(v^{l}\gamma^{*} v^{-l}\otimes v^{l}\gamma v^{-l})\psi \|^2$$ where $v$ is the unitary operator on $H$ such that $v(h_{i,j})=h_{i-1,j}.$
I tried to evaluate this using the inner product but couldn't get a satisfactory result. I will highly appreciate if you can show the way out. Thanks in advance.
