In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space
Contents
- Relationship with other topologies on BH
- Strong operator topology
- Weak star operator topology
- Other properties
- SOT and WOT on BXY when X and Y are normed spaces
- Relationships between different topologies on BXY
- References
Explicitly, for an operator
Equivalently, a net
Relationship with other topologies on B(H)
The WOT is the weakest among all common topologies on
Strong operator topology
The strong operator topology, or SOT, on
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the strong operator topology are precisely those that are continuous in the WOT. Because of this fact, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.
It follows from the polarization identity that a net
Weak-star operator topology
The predual of B(H) is the trace class operators C1(H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H).
A net {Tα} ⊂ B(H) converges to T in WOT if and only Tr(TαF) converges to Tr(TF) for all finite-rank operator F. Since every finite-rank operator is trace-class, this implies that WOT is weaker than the σ-weak topology. To see why the claim is true, recall that every finite-rank operator F is a finite sum F = ∑ λi uivi*. So {Tα} converges to T in WOT means Tr(TαF) = ∑ λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TF).
Extending slightly, one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form S = ∑ λi uivi*, where the series of positive numbers ∑λi converges. Suppose supα ||Tα|| = k < ∞, and Tα converges to T in WOT. For every trace-class S, Tr (TαS) = ∑λi vi*(Tαui) converges to ∑ λi vi*(T ui) = Tr(TS), by invoking, for instance, the dominated convergence theorem.
Therefore every norm-bounded set is compact in WOT, by the Banach–Alaoglu theorem.
Other properties
The adjoint operation T → T*, as an immediate consequence of its definition, is continuous in WOT.
Multiplication is not jointly continuous in WOT: again let
However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net Ti → T in WOT, then STi → ST and TiS → TS in WOT.
SOT and WOT on B(X,Y) when X and Y are normed spaces
We can extend the definitions of SOT and WOT to the more general setting where X and Y are normed spaces and
The strong operator topology on
Relationships between different topologies on B(X,Y)
The different terminology for the various topologies on
In general, the following inclusions hold:
The WOT on