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Strong operator topology

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Title: Strong operator topology  
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Subject: Ultrastrong topology, Weak operator topology, Functional analysis, Weak topology, Strong topology
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Strong operator topology

In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form T\mapsto\|Tx\|, as x varies in H.

Equivalently, it is the coarsest topology such that the evaluation maps T\mapsto Tx (taking values in H) are continuous for any x in H. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon)=\{T:\|Tx-T_0x\|<\epsilon\} (where T0 is any bounded operator on H, x is any vector and ε is any positive real number).

The SOT is stronger than the weak operator topology and weaker than the norm topology.

The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It is more natural too, since it is simply the topology of pointwise convergence for an operator.

The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus.

The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the WOT. Because of this, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.

It should also be noted that the above language translates into convergence properties of Hilbert space operators. One especially observes that for a complex Hilbert space, by way of the polarization identity, one easily verifies that Strong Operator convergence implies Weak Operator convergence.

See also

References

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