Vtyjkyuk

I was reading a paper and encountered the following notation:

Let $mathcalH=ell^2(mathbbZ)$ and $e_p_pin mathbbZ$ be an orthonormal basis of $mathcalH$.Define
$$ue_p=e_p+1quad andquad hatNe_p=pe_p$$
and define the operator
$$W=(1otimes u)^hatNotimes 1=int_mathbbZtimesmathbbTz^sdE_hatN(s)otimes dE_u(z)$$
$$e_kotimes e_lmapsto e_kotimes e_l+k$$ where $dE_hatN$ and $dE_u$ are the spectral measures of $hatN$ 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 $int f(x,y)d(muotimesnu)$ and $int f(x,y)dmuotimes dnu$ are equal or they are completely tow 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.