fa.functional analysis - Direct integrals and fields of operators

Suppose we have a measure space $(X,mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${cal{H}} = int H_x d mu$, which is a Hilbert space.

Suppose now that I have a bounded operator $T$ on $cal H$, about which I know that it is decomposable.

Do you know of any kind of a "formula" which will "compute" a measurable field of operators $T_x$, such that $int T_x =T$?

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