Weak Operator Topology

Name: Weak Operator Topology
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Bol High Quality Content by WIKIPEDIA articles! 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 is continuous for any vectors x and y in the Hilbert space. 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 = ¿ 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.

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High Quality Content by WIKIPEDIA articles! 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 is continuous for any vectors x and y in the Hilbert space. 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 = ¿ 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.

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