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A group of operators, a one-parameter group of operators (cf. [[Operator|Operator]]) on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683701.png" />, i.e. a family of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683703.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683706.png" /> depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683707.png" /> (in the uniform, strong or weak topology). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683708.png" /> is a [[Hilbert space|Hilbert space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683709.png" /> is uniformly bounded, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837010.png" /> is similar to a group of unitary operators (Sz.-Nagy's theorem, cf. also [[Unitary operator|Unitary operator]]).
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A group of operators, a one-parameter group of operators (cf. [[Operator|Operator]]) on a [[Banach space|Banach space]] ,  
 +
i.e. a family of bounded linear operators   U _ {t} ,
 +
$  - \infty < t < \infty $,  
 +
such that $  U _ {0} = I $,  
 +
$  U _ {s+} t = U _ {s} \cdot U _ {t} $
 +
and   U _ {t}
 +
depends continuously on   t (
 +
in the uniform, strong or weak topology). If   E
 +
is a [[Hilbert space|Hilbert space]] and   \| U _ {t} \|
 +
is uniformly bounded, then the group   \{ U _ {t} \}
 +
is similar to a group of unitary operators (Sz.-Nagy's theorem, cf. also [[Unitary operator|Unitary operator]]).
  
 
====References====
 
====References====
Line 6: Line 28:
 
''V.I. Lomonosov''
 
''V.I. Lomonosov''
  
A group with operators, a group with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837012.png" /> is a set of symbols, is a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837013.png" /> such that for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837014.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837015.png" /> there is a corresponding element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837017.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837020.png" /> be groups with the same domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837021.png" />; an isomorphic (a homomorphic) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837023.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837024.png" /> is called an operator isomorphism (operator homomorphism) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837025.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837027.png" />. A subgroup (normal subgroup) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837028.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837029.png" /> with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837030.png" /> is called an admissible subgroup (admissible normal subgroup) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837031.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837032.png" />. The intersection of all admissible subgroups containing a given subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837034.png" /> is called the admissible subgroup generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837035.png" />. A group which does not have admissible normal subgroups apart from itself and the trivial subgroup is called a simple group (with respect to the given domain of operators). Every quotient group of an operator group by an admissible normal subgroup is a group with the same domain of operators.
+
A group with operators, a group with domain of operators   \Sigma ,  
 +
where   \Sigma
 +
is a set of symbols, is a [[Group|group]]   G
 +
such that for every element   a \in G
 +
and every   \sigma \in \Sigma
 +
there is a corresponding element   a \sigma \in G
 +
such that $  ( ab) \sigma = a \sigma \cdot b \sigma $
 +
for any   a, b \in G .  
 +
Let   G
 +
and   G  ^  \prime 
 +
be groups with the same domain of operators   \Sigma ;  
 +
an isomorphic (a homomorphic) mapping   \phi
 +
of   G
 +
onto   G  ^  \prime 
 +
is called an operator isomorphism (operator homomorphism) if $  ( a \sigma ) \phi = ( a \phi ) \sigma $
 +
for any   a \in G ,  
 +
  \sigma \in \Sigma .  
 +
A subgroup (normal subgroup)   H
 +
of the group   G
 +
with domain of operators   \Sigma
 +
is called an admissible subgroup (admissible normal subgroup) if   H \sigma \subseteq H
 +
for any   \sigma \in \Sigma .  
 +
The intersection of all admissible subgroups containing a given subset   M
 +
of   G
 +
is called the admissible subgroup generated by the set   M .  
 +
A group which does not have admissible normal subgroups apart from itself and the trivial subgroup is called a simple group (with respect to the given domain of operators). Every quotient group of an operator group by an admissible normal subgroup is a group with the same domain of operators.
  
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837036.png" /> is called a group with a semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837038.png" /> is a group with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837040.png" /> is a semi-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837041.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837044.png" /> is a semi-group with an identity element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837045.png" />, it is supposed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837046.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837047.png" />. Every group with an arbitrary domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837048.png" /> is a group with semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837050.png" /> is the free semi-group generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837051.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837052.png" /> with semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837053.png" /> possessing an identity element is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837055.png" />-free if it is generated by a system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837056.png" /> such that the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837059.png" />, constitute for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837060.png" /> (as a group without operators) a system of free generators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837061.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837062.png" />-free group (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837063.png" /> being a group of operators), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837064.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837065.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837066.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837067.png" /> be the admissible subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837068.png" /> generated by all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837070.png" />. Then every admissible subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837071.png" /> is an operator free product of groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837072.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837073.png" />-free group (see [[#References|[2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837074.png" /> is a free semi-group of operators, then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837075.png" />, the admissible subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837076.png" />-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837077.png" /> generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837078.png" /> is itself a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837079.png" />-free group with free generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837080.png" /> (see also ).
+
A group   G
 +
is called a group with a semi-group of operators   \Sigma
 +
if   G
 +
is a group with domain of operators   \Sigma ,  
 +
  \Sigma
 +
is a semi-group and $  a( \sigma \tau ) = ( a \sigma ) \tau $
 +
for any   a \in G ,
 +
$  \sigma , \tau \in \Sigma $.  
 +
If   \Sigma
 +
is a semi-group with an identity element   \epsilon ,  
 +
it is supposed that $  a \epsilon = a $
 +
for every   a \in G .  
 +
Every group with an arbitrary domain of operators $  \Sigma _ {0} $
 +
is a group with semi-group of operators   \Sigma ,  
 +
where   \Sigma
 +
is the free semi-group generated by the set $  \Sigma _ {0} $.  
 +
A group   F
 +
with semi-group of operators   \Sigma
 +
possessing an identity element is called   \Sigma -
 +
free if it is generated by a system of elements   X
 +
such that the elements   x \alpha ,  
 +
where   x \in X ,  
 +
  \alpha \in \Sigma ,  
 +
constitute for   F (
 +
as a group without operators) a system of free generators. Let   F
 +
be a   \Gamma -
 +
free group (   \Gamma
 +
being a group of operators), let   \Delta
 +
be a subgroup of   \Gamma ,  
 +
let   f \in F ,  
 +
and let   A _ {f, \Delta } 
 +
be the admissible subgroup of   F
 +
generated by all elements of the form   f ^ { - 1 } ( f \alpha ) ,  
 +
where   \alpha \in \Delta .  
 +
Then every admissible subgroup of   F
 +
is an operator free product of groups of type   A _ {f, \Delta } 
 +
and a   \Gamma -
 +
free group (see [[#References|[2]]]). If   \Sigma
 +
is a free semi-group of operators, then, if   a \neq 1 ,  
 +
the admissible subgroup of the   \Sigma -
 +
free group   F
 +
generated by the element   a
 +
is itself a   \Sigma -
 +
free group with free generator   a (
 +
see also ).
  
An Abelian group with an associative ring of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837081.png" /> is just a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837082.png" />-module (cf. [[Module|Module]]).
+
An Abelian group with an associative ring of operators   K
 +
is just a   K -
 +
module (cf. [[Module|Module]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837083.png" />-free operator groups"  ''Mat. Sb.'' , '''33'''  (1953)  pp. 399–432  (In Russian)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837084.png" />-free operator groups I"  ''Ukr. Mat. Zh.'' , '''16''' :  5  (1964)  pp. 593–602  (In Russian)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837085.png" />-free operator groups II"  ''Ukr. Mat. Zh.'' , '''16''' :  6  (1964)  pp. 730–751  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837083.png" />-free operator groups"  ''Mat. Sb.'' , '''33'''  (1953)  pp. 399–432  (In Russian)</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837084.png" />-free operator groups I"  ''Ukr. Mat. Zh.'' , '''16''' :  5  (1964)  pp. 593–602  (In Russian)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  S.T. Zavalo,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837085.png" />-free operator groups II"  ''Ukr. Mat. Zh.'' , '''16''' :  6  (1964)  pp. 730–751  (In Russian)</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


A group of operators, a one-parameter group of operators (cf. Operator) on a Banach space E , i.e. a family of bounded linear operators U _ {t} , - \infty < t < \infty , such that U _ {0} = I , U _ {s+} t = U _ {s} \cdot U _ {t} and U _ {t} depends continuously on t ( in the uniform, strong or weak topology). If E is a Hilbert space and \| U _ {t} \| is uniformly bounded, then the group \{ U _ {t} \} is similar to a group of unitary operators (Sz.-Nagy's theorem, cf. also Unitary operator).

References

[1] B. Szökevalfi-Nagy, "On uniformly bounded linear transformations in Hilbert space" Acta Sci. Math. (Szeged) , 11 (1947) pp. 152–157
[2] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1948)

V.I. Lomonosov

A group with operators, a group with domain of operators \Sigma , where \Sigma is a set of symbols, is a group G such that for every element a \in G and every \sigma \in \Sigma there is a corresponding element a \sigma \in G such that ( ab) \sigma = a \sigma \cdot b \sigma for any a, b \in G . Let G and G ^ \prime be groups with the same domain of operators \Sigma ; an isomorphic (a homomorphic) mapping \phi of G onto G ^ \prime is called an operator isomorphism (operator homomorphism) if ( a \sigma ) \phi = ( a \phi ) \sigma for any a \in G , \sigma \in \Sigma . A subgroup (normal subgroup) H of the group G with domain of operators \Sigma is called an admissible subgroup (admissible normal subgroup) if H \sigma \subseteq H for any \sigma \in \Sigma . The intersection of all admissible subgroups containing a given subset M of G is called the admissible subgroup generated by the set M . A group which does not have admissible normal subgroups apart from itself and the trivial subgroup is called a simple group (with respect to the given domain of operators). Every quotient group of an operator group by an admissible normal subgroup is a group with the same domain of operators.

A group G is called a group with a semi-group of operators \Sigma if G is a group with domain of operators \Sigma , \Sigma is a semi-group and a( \sigma \tau ) = ( a \sigma ) \tau for any a \in G , \sigma , \tau \in \Sigma . If \Sigma is a semi-group with an identity element \epsilon , it is supposed that a \epsilon = a for every a \in G . Every group with an arbitrary domain of operators \Sigma _ {0} is a group with semi-group of operators \Sigma , where \Sigma is the free semi-group generated by the set \Sigma _ {0} . A group F with semi-group of operators \Sigma possessing an identity element is called \Sigma - free if it is generated by a system of elements X such that the elements x \alpha , where x \in X , \alpha \in \Sigma , constitute for F ( as a group without operators) a system of free generators. Let F be a \Gamma - free group ( \Gamma being a group of operators), let \Delta be a subgroup of \Gamma , let f \in F , and let A _ {f, \Delta } be the admissible subgroup of F generated by all elements of the form f ^ { - 1 } ( f \alpha ) , where \alpha \in \Delta . Then every admissible subgroup of F is an operator free product of groups of type A _ {f, \Delta } and a \Gamma - free group (see [2]). If \Sigma is a free semi-group of operators, then, if a \neq 1 , the admissible subgroup of the \Sigma - free group F generated by the element a is itself a \Sigma - free group with free generator a ( see also ).

An Abelian group with an associative ring of operators K is just a K - module (cf. Module).

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] S.T. Zavalo, "-free operator groups" Mat. Sb. , 33 (1953) pp. 399–432 (In Russian)
[3a] S.T. Zavalo, "-free operator groups I" Ukr. Mat. Zh. , 16 : 5 (1964) pp. 593–602 (In Russian)
[3b] S.T. Zavalo, "-free operator groups II" Ukr. Mat. Zh. , 16 : 6 (1964) pp. 730–751 (In Russian)
How to Cite This Entry:
Operator group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_group&oldid=12052
This article was adapted from an original article by A.P. Mishina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article