# Operator group

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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=48049
This article was adapted from an original article by A.P. Mishina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article