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](http://arxiv.org/abs/0806.0349) and [2011](http://arxiv.org/abs/1005.2656)). 


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!




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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.



definition of operator valued integral with spectral measure
Well, I would make sense of expressions like this by inserting $|e\_n\rangle\langle e\_n|$ between the operator and the measure and then summing over $n$.


Say we want to give a meaning to $A = \int A(x)d\mu$. If we know how to integrate scalar-valued functions against a spectral measure then we can define $$\langle v|A = \sum\_n \int \langle v|A(x)|e\_n\rangle\langle e\_n|d\mu = \sum\_n \langle e\_n|\int \langle v|A(x)|e\_n\rangle d\mu$$ for any $v \in H$, and if we know how to integrate vector-valued functions against scalar measures then we can define $$A|w\rangle = \sum\_n \int A(x)|e\_n\rangle\langle e\_n|d\mu|w\rangle$$ for any $w \in H$. So take your pick, say what the operator is by saying either what happens when you apply it to a vector or what happens when you apply a covector to it.


Do these sums converge? No, not in general, even assuming $A(x)$ is a bounded operator-valued function and $\mu$ is the spectral measure coming from a compact self-adjoint operator. For example, consider the spectral measure coming from the operator of multiplication by $1/n$ on $l^2({\bf N})$. This is a spectral measure defined on $X = \{1/n: n \in {\bf N}\}$ with $\mu(\{1/n\}) = |e\_n\rangle\langle e\_n|$.` Define an operator-valued function on $X$ by setting $A(1/n) = |e\_1\rangle\langle e\_n|$. Then the integral $\int A(x)d\mu$ becomes a sum $\sum\_n |e\_1\rangle\langle e\_n|e\_n\rangle\langle e\_n| = \sum\_n |e\_1\rangle\langle e\_n|$ which does not converge.


Edit: I just looked at their 2011 paper and I found that they have some discussion about how to make their integrals rigorous in Section 2.2. So I think the full answer is that integrals of this form don't always make sense, but there is a heuristic and one can prove convergence in some special cases.


