definition of operator valued integral with spectral measure

Question

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!

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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.

Answer 1

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.