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.

Explicitly, for an operator T there is base of neighborhoods of the following type: choose a finite number of vectors x_i, continuous functionals y_i, and constants ε_i indexed by the same finite set. An operator S lies in the neighborhood if and only if y_i(T(x_i) - S(x_i)) < ε_i for all i.

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 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 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_αS) = ∑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 operator topology
• Topologies on the set of operators on a Hilbert space