Operator group

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)