Weak operator topology

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In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number <Tx, y> is continuous for any vectors x and y in the Hilbert space.

Equivalently, a net TiB(H) of bounded operators converges to TB(H) in WOT if for all y* in H* and x in H, the net y*(Tix) converges to y*(Tx).

Relationship with other topologies on B(H)[edit]

The WOT is the weakest among all common topologies on B(H), the bounded operators on a Hilbert space H.

Strong operator topology[edit]

The strong operator topology, or SOT, on B(H) is the topology of pointwise convergence. Because the inner product is a continuous function, the SOT is stronger than WOT. The following example shows that this inclusion is strict. Let H =  2(N) and consider the sequence {Tn} where T is the unilateral shift. An application of Cauchy-Schwarz shows that Tn → 0 in WOT. But clearly Tn does not converge to 0 in SOT.

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 Tα → 0 in SOT if and only if Tα*Tα → 0 in WOT.

Weak-star operator topology[edit]

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[edit]

The adjoint operation TT*, as an immediate consequence of its definition, is continuous in WOT.

Multiplication is not jointly continuous in WOT: again let T be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both Tn and T*n converges to 0 in WOT. But T*nTn is the identity operator for all n. (Because WOT coincides with the σ-weak topology on bounded sets, multiplication is not jointly continuous in the σ-weak topology.)

However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net TiT in WOT, then STiST and TiSTS in WOT.

See also[edit]