Affiliation:
1. Department of Mathematics, University of South Carolina , Columbia , SC 29208 , United States
Abstract
Abstract
Let
E
E
be a complex Banach lattice and
T
T
is an operator in the center
Z
(
E
)
=
{
T
:
∣
T
∣
≤
λ
I
for some
λ
}
Z\left(E)=\left\{T:| T| \le \lambda I\hspace{0.33em}\hspace{0.1em}\text{for some}\hspace{0.1em}\hspace{0.33em}\lambda \right\}
of
E
E
. Then, the essential norm
‖
T
‖
e
\Vert T{\Vert }_{e}
of
T
T
equals the essential spectral radius
r
e
(
T
)
{r}_{e}\left(T)
of
T
T
. We also prove
r
e
(
T
)
=
max
{
‖
T
A
d
‖
,
r
e
(
T
A
)
}
{r}_{e}\left(T)=\max \left\{\Vert {T}_{}\hspace{-0.35em}{}_{{A}^{d}}\Vert ,{r}_{e}\left({T}_{A})\right\}
, where
T
A
{T}_{A}
is the atomic part of
T
T
and
T
A
d
{T}_{}\hspace{-0.35em}{}_{{A}^{d}}
is the nonatomic part of
T
T
. Moreover,
r
e
(
T
A
)
=
limsup
ℱ
λ
a
{r}_{e}\left({T}_{A})={\mathrm{limsup}}_{{\mathcal{ {\mathcal F} }}}{\lambda }_{a}
, where
ℱ
{\mathcal{ {\mathcal F} }}
is the Fréchet filter on the set
A
A
of all positive atoms in
E
E
of norm one and
λ
a
{\lambda }_{a}
is given by
T
A
a
=
λ
a
a
{T}_{A}a={\lambda }_{a}a
for all
a
∈
A
a\in A
.
Subject
Applied Mathematics,Analysis
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