I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011).

There, they work on a Hilbert space $H$ and on the bounded operators algebra $B(H)$ using some operator valued integrals similar to

$\int_\mathbf{R} A(x)\,dE(x)\;$ and $\;\int_\mathbf{R} dE(x)\,A(x)$,

where $E$ is the spectral resolution of a self-adjoint operator and $A$ is a $B(H)$ valued (norm-continuous) function. I don't know how one defines that.

I don't even know how that one above is defined, since both the measure and the function are operator-valued kinda. What I read about is that you can integrate vector valued functions with respect to a scalar valued measure (Bochner integral or Pettis integral), or scalar valued functions with respect to a spectral resolution (projection valued measure - spectral theorem).

If anyone knows how to define it and/or standard references for it, I would be thankful!

Some further remarks. If this helps, the authors also states that if $E$ is the spectral resolution of the self-adjoint operator $P$, then $P\,dE(p) = p\,dE(p)$, and if $B$ is a compact subset of $\mathbf{R}$ and $F$ is a finite-rank projection in $H$, then by spectral calculus $\int_B A(x)\,F\,dE(x)\;$ and $\;\int_B dE(x)\,F\,A(x)$ are well defined. I can make sense of the second type of integrals, since those will be of finite rank, but not the first type.

Thank you.