Weak operator topology

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2008)

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).

Contents 1 Relationship with other topologies on B(H) 1.1 Strong operator topology 1.2 Weak-star operator topology 2 Other properties 3 See also

[edit] 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.

[edit] 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 = l²(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.

[edit] 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.

[edit] 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.

[edit] See also

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