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 , such that the functional sending an operator to the complex number is continuous for any vectors and in the Hilbert space.

Explicitly, for an operator there is base of neighborhoods of the following type: choose a finite number of vectors , continuous functionals , and constants indexed by the same finite set . An operator lies in the neighborhood if and only if for all .

Equivalently, a net of bounded operators converges to in WOT if for all and , the net converges to .

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

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

Strong operator topology[edit]

The strong operator topology, or SOT, on 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 and consider the sequence of unilateral shifts. An application of Cauchy-Schwarz shows that in WOT. But clearly does not converge to 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 converges to in SOT if and only if 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 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 . (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.

SOT and WOT on B(X,Y) when X and Y are normed spaces[edit]

We can extend the definitions of SOT and WOT to the more general setting where X and Y are normed spaces and is the space of bounded linear operators of the form . In this case, each pair and defines a seminorm on via the rule . The resulting family of seminorms generates the weak operator topology on . Equivalently, the WOT on is formed by taking for basic open neighborhoods those sets of the form , where , is a finite set, is also a finite set, and . The space is a locally convex topological vector space when endowed with the WOT.

The strong operator topology on is generated by the family of seminorms , , via the rules . Thus, a topological base for the SOT is given by open neighborhoods of the form , where as before , is a finite set, and .

Relationships between different topologies on B(X,Y)[edit]

The different terminology for the various topologies on can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is . The weak topology on a normed space is the coarsest topology that makes the linear functionals in continuous; when we take in place of , the weak topology can be very different than the weak operator topology. And while the WOT is formally weaker than the SOT, the SOT is weaker than the operator norm topology.

In general, the following inclusions hold:

and these inclusions may or may not be strict depending on the choices of and .

The WOT on is a formally weaker topology than the SOT, but they nevertheless share some important properties. For example, . Consequently, if is a convex subset of then (in other words, SOT-closure and WOT-closure coincide for convex sets).

See also[edit]