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A topology on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684101.png" /> of continuous linear mappings from one [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684102.png" /> into another topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684103.png" />, converting the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684104.png" /> into a topological vector space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684105.png" /> be a [[Locally convex space|locally convex space]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684106.png" /> be a family of bounded subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684107.png" /> such that the linear hull of the union of the sets of this family is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684108.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o0684109.png" /> be a basis of neighbourhoods of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841010.png" />. The family
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841011.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841012.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841014.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841015.png" />, is a basis of neighbourhoods of zero for a unique topology that is invariant with respect to translation, which is an operator topology and which converts the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841016.png" /> into a locally convex space; this topology is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841017.png" />-topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841018.png" />.
+
A topology on the space  $  L( E, F  ) $
 +
of continuous linear mappings from one [[Topological vector space|topological vector space]]  $  E $
 +
into another topological vector space  $  F $,  
 +
converting the space $  L( E, F  ) $
 +
into a topological vector space. Let  $  F $
 +
be a [[Locally convex space|locally convex space]] and let  $  \mathfrak S $
 +
be a family of bounded subsets of  $  E $
 +
such that the linear hull of the union of the sets of this family is dense in  $  E $.  
 +
Let  $  \mathfrak B $
 +
be a basis of neighbourhoods of zero in  $  F $.  
 +
The family
  
Examples. I) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841019.png" /> be locally convex spaces. 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841020.png" /> be the family of all finite subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841021.png" />; the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841022.png" />-topology (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841023.png" />) is called the topology of simple (or pointwise) convergence. 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841024.png" /> be the family of all convex balanced compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841025.png" />; the corresponding topology is called the topology of convex balanced compact convergence. 3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841026.png" /> be the family of all pre-compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841027.png" />; the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841028.png" />-topology is called the topology of pre-compact convergence. 4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841029.png" /> be the family of all bounded subsets; the corresponding topology is called the topology of bounded convergence.
+
$$
 +
M( S, V) = \{ {f } : {f \in L( E, F  ), f( S) \subset  V } \}
 +
,
 +
$$
  
II) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841030.png" /> are Banach spaces considered simultaneously in the weak or strong (norm) topology, then the corresponding spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841031.png" /> coincide algebraically; the corresponding topologies of simple convergence are called the weak or strong operator topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841032.png" />. The strong operator topology majorizes the weak operator topology; both are compatible with the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841033.png" /> and the space of functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841034.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841038.png" />.
+
where  $  S $
 +
runs through  $  \mathfrak S $
 +
and  $  V $
 +
through  $  \mathfrak B $,  
 +
is a basis of neighbourhoods of zero for a unique topology that is invariant with respect to translation, which is an operator topology and which converts the space $  L( E, F  ) $
 +
into a locally convex space; this topology is called the  $  \mathfrak S $-
 +
topology on $  L( E, F  ) $.
  
III) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841039.png" /> be Hilbert spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841040.png" /> be countable direct sums of the Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841041.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841043.png" /> for all integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841044.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841045.png" /> be the imbedding of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841046.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841047.png" /> defined by the condition that for any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841048.png" /> the restriction of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841049.png" /> to the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841050.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841051.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841052.png" /> and coincides on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841053.png" /> with the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841054.png" />. Then the complete pre-image in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841055.png" /> of the weak (strong) operator topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841056.png" /> is called the ultra-weak (correspondingly, ultra-strong) operator topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841057.png" />. The ultra-weak (ultra-strong) topology majorizes the weak (strong) operator topology. A symmetric subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841058.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841059.png" /> of all bounded linear operators on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841060.png" />, containing the identity operator, coincides with the set of all operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841061.png" /> that commute with each operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841062.png" /> that commutes with all operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841063.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841064.png" /> is closed in the weak (or strong, or ultra-weak, or ultra-strong) operator topology, i.e. is a [[Von Neumann algebra|von Neumann algebra]].
+
Examples. I) Let $  E, F $
 +
be locally convex spaces. 1) Let  $  \mathfrak S $
 +
be the family of all finite subsets in  $  E $;
 +
the corresponding  $  \mathfrak S $-
 +
topology (on  $  L( E, F  ) $)
 +
is called the topology of simple (or pointwise) convergence. 2) Let  $  \mathfrak S $
 +
be the family of all convex balanced compact subsets of  $  E $;
 +
the corresponding topology is called the topology of convex balanced compact convergence. 3) Let  $  \mathfrak S $
 +
be the family of all pre-compact subsets of  $  E $;
 +
the corresponding  $  \mathfrak S $-
 +
topology is called the topology of pre-compact convergence. 4) Let  $  \mathfrak S $
 +
be the family of all bounded subsets; the corresponding topology is called the topology of bounded convergence.
 +
 
 +
II) If  $  E, F $
 +
are Banach spaces considered simultaneously in the weak or strong (norm) topology, then the corresponding spaces  $  L( E, F  ) $
 +
coincide algebraically; the corresponding topologies of simple convergence are called the weak or strong operator topologies on  $  L( E, F  ) $.
 +
The strong operator topology majorizes the weak operator topology; both are compatible with the duality between  $  L( E, F  ) $
 +
and the space of functionals on  $  L( E, F  ) $
 +
of the form  $  f( A) = \sum \phi _ {i} ( A \xi _ {i} ) $,
 +
where  $  \xi _ {i} \in E $,
 +
$  \phi _ {i} \in F ^ { * } $,
 +
$  A \in L( E, F  ) $.
 +
 
 +
III) Let  $  E, F $
 +
be Hilbert spaces and let $  \widetilde{E}  , \widetilde{F}  $
 +
be countable direct sums of the Hilbert spaces $  E _ {n} , F _ {n} $,  
 +
respectively, where $  E _ {n} = E $,  
 +
$  F _ {n} = F $
 +
for all integer $  n $;  
 +
let $  \psi $
 +
be the imbedding of the space $  L( E, F  ) $
 +
into $  L( \widetilde{E}  , \widetilde{F}  ) $
 +
defined by the condition that for any operator $  A \in L( E, F  ) $
 +
the restriction of the operator $  \psi ( A) $
 +
to the subspace $  E _ {n} $
 +
maps $  E _ {n} $
 +
into $  F _ {n} $
 +
and coincides on $  E _ {n} $
 +
with the operator $  A $.  
 +
Then the complete pre-image in $  L( E, F  ) $
 +
of the weak (strong) operator topology on $  L( \widetilde{E}  , \widetilde{F}  ) $
 +
is called the ultra-weak (correspondingly, ultra-strong) operator topology on $  L( E, F  ) $.  
 +
The ultra-weak (ultra-strong) topology majorizes the weak (strong) operator topology. A symmetric subalgebra $  \mathfrak A $
 +
of the algebra $  L( E) $
 +
of all bounded linear operators on a Hilbert space $  E $,  
 +
containing the identity operator, coincides with the set of all operators from $  L( E) $
 +
that commute with each operator from $  L( E) $
 +
that commutes with all operators from $  \mathfrak A $,  
 +
if and only if $  \mathfrak A $
 +
is closed in the weak (or strong, or ultra-weak, or ultra-strong) operator topology, i.e. is a [[Von Neumann algebra|von Neumann algebra]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sakai,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841065.png" />-algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841066.png" />-algebras" , Springer  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sakai,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841065.png" />-algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068410/o06841066.png" />-algebras" , Springer  (1971)</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


A topology on the space $ L( E, F ) $ of continuous linear mappings from one topological vector space $ E $ into another topological vector space $ F $, converting the space $ L( E, F ) $ into a topological vector space. Let $ F $ be a locally convex space and let $ \mathfrak S $ be a family of bounded subsets of $ E $ such that the linear hull of the union of the sets of this family is dense in $ E $. Let $ \mathfrak B $ be a basis of neighbourhoods of zero in $ F $. The family

$$ M( S, V) = \{ {f } : {f \in L( E, F ), f( S) \subset V } \} , $$

where $ S $ runs through $ \mathfrak S $ and $ V $ through $ \mathfrak B $, is a basis of neighbourhoods of zero for a unique topology that is invariant with respect to translation, which is an operator topology and which converts the space $ L( E, F ) $ into a locally convex space; this topology is called the $ \mathfrak S $- topology on $ L( E, F ) $.

Examples. I) Let $ E, F $ be locally convex spaces. 1) Let $ \mathfrak S $ be the family of all finite subsets in $ E $; the corresponding $ \mathfrak S $- topology (on $ L( E, F ) $) is called the topology of simple (or pointwise) convergence. 2) Let $ \mathfrak S $ be the family of all convex balanced compact subsets of $ E $; the corresponding topology is called the topology of convex balanced compact convergence. 3) Let $ \mathfrak S $ be the family of all pre-compact subsets of $ E $; the corresponding $ \mathfrak S $- topology is called the topology of pre-compact convergence. 4) Let $ \mathfrak S $ be the family of all bounded subsets; the corresponding topology is called the topology of bounded convergence.

II) If $ E, F $ are Banach spaces considered simultaneously in the weak or strong (norm) topology, then the corresponding spaces $ L( E, F ) $ coincide algebraically; the corresponding topologies of simple convergence are called the weak or strong operator topologies on $ L( E, F ) $. The strong operator topology majorizes the weak operator topology; both are compatible with the duality between $ L( E, F ) $ and the space of functionals on $ L( E, F ) $ of the form $ f( A) = \sum \phi _ {i} ( A \xi _ {i} ) $, where $ \xi _ {i} \in E $, $ \phi _ {i} \in F ^ { * } $, $ A \in L( E, F ) $.

III) Let $ E, F $ be Hilbert spaces and let $ \widetilde{E} , \widetilde{F} $ be countable direct sums of the Hilbert spaces $ E _ {n} , F _ {n} $, respectively, where $ E _ {n} = E $, $ F _ {n} = F $ for all integer $ n $; let $ \psi $ be the imbedding of the space $ L( E, F ) $ into $ L( \widetilde{E} , \widetilde{F} ) $ defined by the condition that for any operator $ A \in L( E, F ) $ the restriction of the operator $ \psi ( A) $ to the subspace $ E _ {n} $ maps $ E _ {n} $ into $ F _ {n} $ and coincides on $ E _ {n} $ with the operator $ A $. Then the complete pre-image in $ L( E, F ) $ of the weak (strong) operator topology on $ L( \widetilde{E} , \widetilde{F} ) $ is called the ultra-weak (correspondingly, ultra-strong) operator topology on $ L( E, F ) $. The ultra-weak (ultra-strong) topology majorizes the weak (strong) operator topology. A symmetric subalgebra $ \mathfrak A $ of the algebra $ L( E) $ of all bounded linear operators on a Hilbert space $ E $, containing the identity operator, coincides with the set of all operators from $ L( E) $ that commute with each operator from $ L( E) $ that commutes with all operators from $ \mathfrak A $, if and only if $ \mathfrak A $ is closed in the weak (or strong, or ultra-weak, or ultra-strong) operator topology, i.e. is a von Neumann algebra.

References

[1] H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[2] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988)
[3] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[4] S. Sakai, "-algebras and -algebras" , Springer (1971)
How to Cite This Entry:
Operator topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_topology&oldid=12370
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article