Affiliation:
1. Department of Mathematics and Computer Science, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University , Fez , Morocco
Abstract
Abstract
Let
X
X
and
Y
Y
be non-Archimedean Banach spaces over
K
{\mathbb{K}}
,
A
∈
B
(
X
,
Y
)
A\in B\left(X,Y)
and
B
∈
B
(
Y
,
X
)
B\in B\left(Y,X)
such that
A
B
A
=
A
2
ABA={A}^{2}
and
B
A
B
=
B
2
.
BAB={B}^{2}.
In this article, we investigate some properties of the operator equations
A
B
A
=
A
2
ABA={A}^{2}
and
B
A
B
=
B
2
BAB={B}^{2}
, and many common basic properties of
I
Y
−
A
B
{I}_{Y}-AB
and
I
X
−
B
A
{I}_{X}-BA
are given. In particular, if
X
X
and
Y
Y
are Banach spaces over a spherically complete field
K
,
{\mathbb{K}},
then
N
(
I
Y
−
A
B
)
N\left({I}_{Y}-AB)
is a complemented subspace of
Y
Y
if and only if
N
(
I
X
−
B
A
)
N\left({I}_{X}-BA)
is a complemented subspace of
X
.
X.
Finally, we give some examples to illustrate our work.
Subject
Applied Mathematics,Geometry and Topology,Algebra and Number Theory