# definition of operator valued integral with spectral measure

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 integral similar to

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

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.

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), which is the core of the spectral theorem.

I, however, don't even know how that one above is defined, since both the measure and the function are operator-valued kinda.

If anyone knows how to define it and where to look for it, I would be thankful! Cheers.

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If $A(x)$ where simple, that is, of the form $\sum_{j=1}^N\chi_{S_j}T_j$, where $S_j$ are Borel measurable sets and $T_j$ bounded operators, then a natural definition would be $$\int_{\Bbb R}A(x)dE(x):=\sum_{j=1}^NT_jE(S_j).$$ – Davide Giraudo Jan 4 '13 at 22:27
Take a look at the discussion here of integration against spectral measures: planetmath.org/encyclopedia/… – Branimir Ćaćić Jan 4 '13 at 22:31
@Nobert, thank you for your note, but that question is about the spectral resolution of the multiplication operator in $L^2$, quite not the problem described here. – Yul Otani Jan 4 '13 at 22:53
@YulOtani: Gah! Yes, you're right. The only thing that comes to my mind is to approximate in the strong topology by the integrals of simple functions mentioned by Davide Giraudo. – Branimir Ćaćić Jan 4 '13 at 23:32
Also asked on MO. – Martin Jan 6 '13 at 22:08