Operator power of another operator

Question

I was reading a paper and encountered the following notation:

Let $\mathcal{H}=\ell^2(\mathbb{Z})$ and $\{e_p\}_{p\in \mathbb{Z}}$ be an orthonormal basis of $\mathcal{H}$.Define $$ue_p=e_{p+1}\quad and\quad \hat{N}e_p=pe_p$$ and define the operator $$W=(1\otimes u)^{\hat{N}\otimes 1}=\int_{\mathbb{Z}\times\mathbb{T}}z^sdE_{\hat{N}}(s)\otimes dE_{u}(z)$$ $$e_k\otimes e_l\mapsto e_k\otimes e_{l+k}$$ where $dE_{\hat{N}}$ and $dE_{u}$ are the spectral measures of $\hat{N}$ and $u$, respectively.

I know how to integrate with respect to a single measure or with respect to product measures. But here, the integration is with respect to tensor product of measures. I am confused whether $\int f(x,y)d(\mu\otimes\nu)$ and $\int f(x,y)d\mu\otimes d\nu$ are equal or they are completely two different things. I guess they are not equal.

Can anyone please elaborate what is going on in the definition of $W$ and how to calculate this type of integrals? or are there any other sense in which one can give meaning to an operator raised to the power of another operator? Any good reference in this direction would be highly appreciated.

Thanx in advance.

Answer 1

If $\mu$ and $\nu$ are, respectively, projection valued measures on $X$ and $Y$, taking values in $B(H)$, then $\mu \otimes \nu$ can be defined to be a projection valued measure on $X\times Y$ taking values in $B(H)\otimes B(H)$. I don't see anything more than that going on here.

In answer to the general question about operator powers, it looks like what is being done here is that we have two commuting normal operators ($1\otimes u$ and $\hat{N}\otimes 1$), so they can be simultaneously realized as, say, multiplication by $f$ and $g$, and the power is taken to be multiplication by $f^g$, which is well-defined if $g$ is integer-valued and $f$ doesn't vanish.