Direct integral of operators - Imported Knowlpedia Content

Definition

Let $\{H_x\}_{x\in X}$ be a measurable field of Hilbert spaces over $(X,\mu)$ , and let $A_x$ be a measurable family of closed densely defined operators on the fibers. The direct integral

A=\int_X^\oplus A_x\,d\mu(x)

has domain consisting of measurable sections $\xi$ such that $\xi(x)\in\operatorname{Dom}(A_x)$ almost everywhere and $x\mapsto A_x\xi(x)$ is square-integrable; it acts by $(A\xi)(x)=A_x\xi(x)$ . Here measurability of the family can be expressed by measurability of the graph projections. Operators and families that agree almost everywhere define the same direct integral.

Bounded and unbounded cases

If the $A_x$ are bounded and $\operatorname*{ess\,sup}_x\lVert A_x\rVert<\infty$ , then $A$ is the bounded decomposable operator determined by the field. Without a uniform essential bound, the same formula usually defines an unbounded operator with the stated maximal natural domain. The graph of $A$ is the direct integral of the fiber graphs, so closedness passes from the fibers to $A$ .

Fiberwise spectral properties

Adjoints and standard operator properties are computed fiberwise under the measurability hypotheses:

A^*=\int_X^\oplus A_x^*\,d\mu(x).

In particular, $A$ is self-adjoint when $A_x$ is self-adjoint almost everywhere. Fiberwise functional calculus then gives

f(A)=\int_X^\oplus f(A_x)\,d\mu(x)

for bounded Borel $f$ . These results are part of the direct-integral operator theory in Takesaki, Chapter IV, §8.

Examples and scope

For one-dimensional fibers $H_x=\mathbb C$ , the direct integral of scalar operators $A_x z=a(x)z$ is the multiplication operator by the measurable function $a$ on $L^2(X,\mu)$ , possibly unbounded. A bare family of closed operators need not be measurable and therefore need not define a direct integral.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV, §8, direct integrals and decomposable operators.
Jacques Dixmier, Von Neumann Algebras, North-Holland, 1981. Publisher record. Relevant: Chapter II, measurable fields and direct integrals of operators.