Affiliation:
1. Department of Mathematics , Shiv Nadar University , Delhi (NCR) , India
Abstract
Abstract
Let
X
,
Y
{X,Y}
be real, infinite-dimensional Banach spaces. Let
ℒ
(
X
,
Y
)
{{\mathcal{L}}(X,Y)}
be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at
A
∈
ℒ
(
X
,
Y
)
{A\in{\mathcal{L}}(X,Y)}
is to estimate the limit
(which always exists)
lim
t
→
0
+
∥
A
+
t
B
∥
-
∥
A
∥
t
for
B
∈
ℒ
(
X
,
Y
)
,
\lim_{t\rightarrow 0^{+}}\frac{\lVert A+tB\rVert-\lVert A\rVert}{t}\quad\text{%
for }B\in{\mathcal{L}}(X,Y),
using the values of B on the state space
S
A
=
{
τ
∈
ℒ
(
X
,
Y
)
∗
:
τ
(
A
)
=
∥
A
∥
,
∥
τ
∥
=
1
}
.
S_{A}=\bigl{\{}\tau\in{\mathcal{L}}(X,Y)^{\ast}:\tau(A)=\lVert A\rVert,\,%
\lVert\tau\rVert=1\bigr{\}}.
In this paper, we give several examples of Banach spaces, including the
ℓ
p
{\ell^{p}}
spaces
(for
1
<
p
<
∞
{1<p<\infty}
)
where a more tangible estimate is possible, under additional hypotheses on A.
We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this.
Our results also show that the operator subdifferential limit is related to the
corresponding subdifferential limit of the vectors in the range space, when
A
∗
∗
{A^{\ast\ast}}
attains its norm.
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics, Probability and Uncertainty,Mathematical Physics
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1. Subdifferentiability and polyhedrality of the norm;Bollettino dell'Unione Matematica Italiana;2023-04-21