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EspañolWEAK 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 ''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'').
The WOT is the weakest among all common topologies on ''B'' (''H''), the bounded operators on a Hilbert space ''H''.
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 ''Tα'' → 0 in SOT if and only if ''Tα
★ Tα'' → 0 in WOT.
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 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.
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.
★ Weak topology
★ Weak-star topology
★ Topologies on the set of operators on a 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'').
| Contents |
| Relationship with other topologies on ''B''(''H'') |
| Strong operator topology |
| Weak-star operator topology |
| Other properties |
| See also |
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 ''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 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 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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