Operator group

A group of operators, a one-parameter group of operators (cf. Operator) on a Banach space , i.e. a family of bounded linear operators , , such that , and depends continuously on (in the uniform, strong or weak topology). If is a Hilbert space and is uniformly bounded, then the group 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 , where is a set of symbols, is a group such that for every element and every there is a corresponding element such that for any . Let and be groups with the same domain of operators ; an isomorphic (a homomorphic) mapping of onto is called an operator isomorphism (operator homomorphism) if for any , . A subgroup (normal subgroup) of the group with domain of operators is called an admissible subgroup (admissible normal subgroup) if for any . The intersection of all admissible subgroups containing a given subset of is called the admissible subgroup generated by the set . 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 is called a group with a semi-group of operators if is a group with domain of operators , is a semi-group and for any , . If is a semi-group with an identity element , it is supposed that for every . Every group with an arbitrary domain of operators is a group with semi-group of operators , where is the free semi-group generated by the set . A group with semi-group of operators possessing an identity element is called -free if it is generated by a system of elements such that the elements , where , , constitute for (as a group without operators) a system of free generators. Let be a -free group ( being a group of operators), let be a subgroup of , let , and let be the admissible subgroup of generated by all elements of the form , where . Then every admissible subgroup of is an operator free product of groups of type and a -free group (see [2]). If is a free semi-group of operators, then, if , the admissible subgroup of the -free group generated by the element is itself a -free group with free generator (see also ).

An Abelian group with an associative ring of operators is just a -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)