Existence of inverses and square roots in locally Banach semigroups with identity
Authors: Eslinger Robert C. 
Published: June 2010

Let

S
S

be a multiplicative topological semigroup with identity

e
e

. Suppose

D
D

is an open subset containing

e
e

and

h
h

is a homeomorphism from

D
D

onto a Banach space

B
B

with

h
(
e
)
=
0

h(e) = 0

. Define the function

P
P

by

P
(
x
,
y
)
=
h
[

h

−
1

(
x
)
⋅

h

−
1

(
y
)
]

P(x,y) = h[{h^{ - 1}}(x) \cdot {h^{ - 1}}(y)]

. A new implicit function theorem is applied to the function

P
P

to show the existence of inverses and square roots of elements in a neighborhood of the identity. It is assumed that

P
P

satisfies the following condition: There exist a one-one function

A
A

from a subset of

B
B

into

B
B

and positive numbers

r
,
M

r,M

, and

c
c

such that (i) if

|

|

x

|

|

>
r

||x|| > r

then

x
∈
dom
⁡
(

A

−
1

)

x \in \operatorname {dom} ({A^{ - 1}})

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper A Superscript negative 1 Baseline left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals upper M StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue">

|

|

A

−
1

(
x
)

|

|

⩽
M

|

|

x

|

|

||{A^{ - 1}}(x)|| \leqslant M||x||

, (ii)

c
M
>
1

cM > 1

, and (iii) if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue x Subscript i Baseline StartAbsoluteValue EndAbsoluteValue comma StartAbsoluteValue EndAbsoluteValue y Subscript i Baseline StartAbsoluteValue EndAbsoluteValue greater-than r left-parenthesis i equals 1 comma 2 right-parenthesis">

|

|

x
i

|

|

,

|

|

y
i

|

|

>
r
(
i
=
1
,
2
)

||{x_i}||,||{y_i}|| > r(i = 1,2)

then

(

x
i

,

y
j

)
∈
dom
⁡
(
P
)
(
i
,
j
=
1
,
2
)

({x_i},{y_j}) \in \operatorname {dom} (P)(i,j = 1,2)

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper P left-parenthesis x 1 comma y 1 right-parenthesis minus upper P left-parenthesis x 2 comma y 2 right-parenthesis minus upper A left-parenthesis y 1 minus y 2 right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals c StartAbsoluteValue EndAbsoluteValue y 1 minus y 2 StartAbsoluteValue EndAbsoluteValue comma">

|

|

P
(

x
1

,

y
1

)
−
P
(

x
2

,

y
2

)
−
A
(

y
1

−

y
2

)

|

|

⩽
c

|

|

y
1

−

y
2

|

|

,

||P({x_1},{y_1}) - P({x_2},{y_2}) - A({y_1} - {y_2})|| \leqslant c||{y_1} - {y_2}||,

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper P left-parenthesis x 1 comma y 1 right-parenthesis minus upper P left-parenthesis x 2 comma y 1 right-parenthesis minus upper A left-parenthesis x 1 minus x 2 right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals c StartAbsoluteValue EndAbsoluteValue x 1 minus x 2 StartAbsoluteValue EndAbsoluteValue period">

|

|

P
(

x
1

,

y
1

)
−
P
(

x
2

,

y
1

)
−
A
(

x
1

−

x
2

)

|

|

⩽
c

|

|

x
1

−

x
2

|

|

.

||P({x_1},{y_1}) - P({x_2},{y_1}) - A({x_1} - {x_2})|| \leqslant c||{x_1} - {x_2}||.

\]