Difference between revisions of "Operator group"
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− | A group of operators, a one-parameter group of operators (cf. [[Operator|Operator]]) on a [[Banach space|Banach space]] | + | <!-- |
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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==== | ||
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''V.I. Lomonosov'' | ''V.I. Lomonosov'' | ||
− | A group with operators, a group with 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 | + | 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 | + | 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, "![]() |
[3a] | S.T. Zavalo, "![]() |
[3b] | S.T. Zavalo, "![]() |
Operator group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_group&oldid=12052