Weak operator topology

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 "T_i" ⊂ "B"("H") of bounded operators converges to "T" ∈ "B"("H") in WOT if for all "y*" in "H*" and "x" in "H", the net "y*"("T_ix") converges to "y*"("Tx").

Relationship with other topologies on "B"("H")

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

Strong operator topology

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" = "l"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 which are continuous in the strong operator topology are precisely those which 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 "T_α" → 0 in SOT if and only if "T_α*T_α" → 0 in WOT.

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 {"T_α"} ⊂ "B"("H") converges to "T" in WOT if and only Tr("T_αF") 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" = ∑ "λ_i u_iv_i*". So {"T_α"} converges to "T" in WOT means Tr("T_αF") = ∑ "λ_i v_i*"("T_αu_i") converges to ∑ "λ_i v_i*"("T u_i") = 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" = ∑ "λ_i u_iv_i*", where the series of positive numbers ∑"λ_i" converges. Suppose sup_"α" ||"T_α"|| = "k" < ∞, and "T_α" converges to "T" in WOT. For every trace-class "S", Tr ("T_α"S) = ∑"λ_i v_i*"("T_αu_i") converges to ∑ "λ_i v_i*"("T u_i") = 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 "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 "T_i" → "T" in WOT, then "ST_i" → "ST" and "T_iS" → "TS" in WOT.

See also

*Weak topology
*Weak-star topology
*Topologies on the set of operators on a Hilbert space

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