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The article page says "... SOT, is the weakest 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 \|Tx\| is continuous for each vector x in the Hilbert space."
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So defined topology is not translation invariant, nor locally convex, nor even
Hausdorff, nor satisfies most claims of the article. For example it is not Hausdorff since there is no open set distinguishing the right shift S and its power
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Please, correct the article (provisional correction done by adding 'locally convex' to include the translational invariance ad hock). Remember that no information is much better than wrong information.
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I removed the text (example) below from the article, since it is affected by correcting the definition. First to be said, the example claims to show
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615:——————— As an example of this lack of nicer properties, let us mention that the involution map is not continuous in this topology: fix an
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Copy the definition correctly from the source, usually no word (such as 'locally convex' for example) can be ommited.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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in the SOT topology. This means that the adjoint operation is not SOT-continuous. ————————
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There are other uses for SOT as an acronym. e.g. Small
Outline Transistor. Sound on tape.
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converges in SOT to identity (while the adjoints do not), supporting it by formula
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204:(Note that also the example turned affected by correcting the definition.)
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is not continuous, for which it (silently) argues (or it seems to) that
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SOT as an acronym should be a disambiguation page, not a redirect.
891:{\displaystyle S^{*}(e_{n})=e_{n-1},n\geq 1,\ \ S^{*}(e_{0})=0.}
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1073:{\displaystyle \lim _{n\rightarrow \infty }(S^{*})^{n}=0}
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Please would you check that and reformulate the example?
507:{\displaystyle \lim _{n\rightarrow \infty }(S^{*})^{n}=0}
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I corrected the definition and added an equivalent one.
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After the correction of definition, we would need that
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658:{\displaystyle \{e_{n}:n\in \mathbb {N} \}}
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517:in the SOT topology while its adjoints
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665:of a Hilbert space and consider the
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95:This article is within the scope of
38:It is of interest to the following
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984:{\displaystyle \|S^{n}(x)\|=\|x\|}
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348:{\displaystyle \|S^{n}(x)\|=\|x\|}
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1120:Low-priority mathematics articles
742:{\displaystyle S(e_{n})=e_{n+1}.}
115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
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403:{\displaystyle \|S^{n}(x)-x\|}
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233:11:04, 16 September 2014 (UTC)
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109:and see a list of open tasks.
1115:C-Class mathematics articles
238:The example about * removed
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927:{\displaystyle \{S^{n}\}}
216:16:44, 2 April 2010 (UTC)
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291:{\displaystyle S^{n}(x)}
158:The definition is wrong.
141:project's priority scale
439:{\displaystyle x=e_{1}}
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775:{\displaystyle S^{*}}
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187:{\displaystyle S^{2}}
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62:Low‑priority
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752:Then the
688:given by
225:Kappaenne
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139:on the
30:C-class
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36:scale.
1097:talk
229:talk
212:talk
1093:DFH
1027:lim
608:.
461:lim
131:Low
1111::
1099:)
1050:∗
1037:∞
1034:→
979:‖
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967:‖
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886:0.
862:∗
844:≥
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645:∈
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471:∞
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1095:(
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958:(
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803:(
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737:.
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702:(
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676:S
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649:N
642:n
639::
634:n
630:e
626:{
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593:=
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528:(
525:(
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476:(
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424:=
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395:x
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383:(
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325:x
322:(
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313:S
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280:(
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271:S
227:(
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42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
