- Weak operator topology
In
functional analysis , the weak operator topology, often abbreviated WOT, is the weakesttopology on the set ofbounded operator s on aHilbert 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 "Ti" ⊂ "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*"("Tix") 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 functional s on the set of bounded operators on a Hilbert space which are continuous in thestrong operator topology are precisely those which are continuous in the WOT. Because of this fact, the closure of aconvex 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
The predual of "B"("H") is the
trace class operators C1("H"), and it generates the w*-topology on "B"("H"), called theweak-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 sup"α" ||"Tα"|| = "k" < ∞, and "Tα" converges to "T" in WOT. For every trace-class "S", Tr ("Tα"S) = ∑"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
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 "Ti" → "T" in WOT, then "STi" → "ST" and "TiS" → "TS" in WOT.
See also
*
Weak topology
*Weak-star topology
*Topologies on the set of operators on a Hilbert space
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