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− | A group of operators, a one-parameter group of operators (cf. [[Operator|Operator]]) on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683701.png" />, i.e. a family of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683703.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683706.png" /> depends continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683707.png" /> (in the uniform, strong or weak topology). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683708.png" /> is a [[Hilbert space|Hilbert space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o0683709.png" /> is uniformly bounded, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837010.png" /> is similar to a group of unitary operators (Sz.-Nagy's theorem, cf. also [[Unitary operator|Unitary operator]]). | + | <!-- |
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| + | A group of operators, a one-parameter group of operators (cf. [[Operator|Operator]]) on a [[Banach space|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|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==== |
Line 6: |
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| ''V.I. Lomonosov'' | | ''V.I. Lomonosov'' |
| | | |
− | A group with operators, a group with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837012.png" /> is a set of symbols, is a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837013.png" /> such that for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837014.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837015.png" /> there is a corresponding element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837017.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837020.png" /> be groups with the same domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837021.png" />; an isomorphic (a homomorphic) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837023.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837024.png" /> is called an operator isomorphism (operator homomorphism) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837025.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837027.png" />. A subgroup (normal subgroup) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837028.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837029.png" /> with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837030.png" /> is called an admissible subgroup (admissible normal subgroup) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837031.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837032.png" />. The intersection of all admissible subgroups containing a given subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837034.png" /> is called the admissible subgroup generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837035.png" />. 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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837036.png" /> is called a group with a semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837038.png" /> is a group with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837040.png" /> is a semi-group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837041.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837044.png" /> is a semi-group with an identity element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837045.png" />, it is supposed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837046.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837047.png" />. Every group with an arbitrary domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837048.png" /> is a group with semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837050.png" /> is the free semi-group generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837051.png" />. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837052.png" /> with semi-group of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837053.png" /> possessing an identity element is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837055.png" />-free if it is generated by a system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837056.png" /> such that the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837059.png" />, constitute for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837060.png" /> (as a group without operators) a system of free generators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837061.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837062.png" />-free group (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837063.png" /> being a group of operators), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837064.png" /> be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837065.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837066.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837067.png" /> be the admissible subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837068.png" /> generated by all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837070.png" />. Then every admissible subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837071.png" /> is an operator free product of groups of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837072.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837073.png" />-free group (see [[#References|[2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837074.png" /> is a free semi-group of operators, then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837075.png" />, the admissible subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837076.png" />-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837077.png" /> generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837078.png" /> is itself a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837079.png" />-free group with free generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837080.png" /> (see also ). | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837081.png" /> is just a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068370/o06837082.png" />-module (cf. [[Module|Module]]). | + | 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> |
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) |