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In [[functional analysis]], a branch of [[mathematics]], the '''strong operator topology''', often abbreviated SOT, is the weakest [[locally convex]] [[topology]] on the set of [[bounded operator]]s on a [[Hilbert space]] (or, more generally, on a [[Banach space]]) such that the evaluation map sending an operator ''T'' to the real number <math>\|Tx\|</math> is [[continuous function (topology)|continuous]] for each vector ''x'' in the Hilbert space. | |||
The SOT is [[finer topology|stronger]] than the [[weak operator topology]] and weaker than the [[operator norm|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 functional]]s on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the [[weak operator topology|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== | |||
*[[Strongly continuous semigroup]] | |||
*[[Topologies on the set of operators on a Hilbert space]] | |||
==References== | |||
*{{cite book |last=Rudin |first=Walter |title=Functional Analysis |date=January 1991 |publisher=McGraw-Hill Science/Engineering/Math |isbn=0-07-054236-8}} | |||
*{{cite book |last=Pedersen |first=Gert |title=Analysis Now |year=1989 |publisher=Springer |isbn=0-387-96788-5}} | |||
{{Functional Analysis}} | |||
[[Category:Topology of function spaces]] | |||
Revision as of 00:07, 14 September 2013
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space (or, more generally, on a Banach space) such that the evaluation map sending an operator T to the real number is continuous for each vector x in the Hilbert space.
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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