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, "![]() |
[3a] | S.T. Zavalo, "![]() |
[3b] | S.T. Zavalo, "![]() |
Operator group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_group&oldid=48049