Decomposable operator - Imported Knowlpedia Content

Definition

Let $\mathcal H=\int_X^\oplus H_x\,d\mu(x)$ and $\mathcal K=\int_X^\oplus K_x\,d\mu(x)$ be direct integrals of [[linear-algebra/hilbert-space|Hilbert spaces]]. A bounded operator $T:\mathcal H\to\mathcal K$ is decomposable if there is a measurable field of bounded operators $x\mapsto T_x:H_x\to K_x$ , with $\operatorname*{ess\,sup}_x\lVert T_x\rVert<\infty$ , such that

(T\xi)(x)=T_x\xi(x)

for almost every $x$ and every measurable square-integrable section $\xi$ . One writes $T=\int_X^\oplus T_x\,d\mu(x)$ . The field $T_x$ is determined up to equality almost everywhere, and $\lVert T\rVert=\operatorname*{ess\,sup}_x\lVert T_x\rVert$ .

Algebraic operations

Sums, products of composable decomposable operators, and adjoints are decomposable and are computed fiberwise:

(ST)_x=S_xT_x,\qquad (T^*)_x=T_x^*

almost everywhere. Consequently, decomposable operators on a fixed direct integral form a von Neumann algebra. Fiberwise properties such as self-adjointness, positivity, or unitarity pass to the global operator and back, modulo null sets.

Characterization by diagonal operators

For $f\in L^\infty(X,\mu)$ , let $M_f$ act diagonally by $(M_f\xi)(x)=f(x)\xi(x)$ . Under the standard separability and $\sigma$ -finiteness hypotheses used in direct-integral theory, the decomposable operators are precisely the bounded operators commuting with every $M_f$ . Thus decomposability is an intrinsic commutant condition, not a claim that the fibers $T_x$ are diagonalizable Takesaki, Chapter IV, §8.

Examples and scope

A diagonal multiplication operator is decomposable with $T_x=f(x)I_{H_x}$ . More generally, a measurable essentially bounded family of matrices defines a decomposable operator on a direct integral of finite-dimensional fibers. An operator that mixes values at different base points, such as translation on $L^2(\mathbb R)$ , is generally not decomposable over the usual position-space decomposition.

References

Jacques Dixmier, Von Neumann Algebras, North-Holland, 1981. Publisher record. Relevant: Chapter II, §2 on decomposable operators.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV, §8 on direct integrals and decomposable operators.