Abstract
Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$
(
Ω
,
Σ
,
λ
)
such that $$L^\infty \subset E\subset L^1$$
L
∞
⊂
E
⊂
L
1
. Let $$E'$$
E
′
denote the Köthe dual of E and $$\tau (E,E')$$
τ
(
E
,
E
′
)
stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$
T
:
E
→
X
between the locally convex space $$(E,\tau (E,E'))$$
(
E
,
τ
(
E
,
E
′
)
)
and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$
T
:
L
∞
→
X
between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$
(
L
∞
,
τ
(
L
∞
,
L
1
)
)
and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$
m
T
:
Σ
→
X
has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$
|
m
T
|
(
Ω
)
=
‖
T
‖
nuc
(= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$
L
∞
are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$
T
:
L
∞
→
X
admits a factorization through some Orlicz space $$L^\varphi $$
L
φ
, that is, $$T=S\circ i_\infty $$
T
=
S
∘
i
∞
, where $$S:L^\varphi \rightarrow X$$
S
:
L
φ
→
X
is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$
i
∞
:
L
∞
→
L
φ
is the inclusion map.
A Criterion for Boundedness of a Linear Map from any Banach Space into a Banach Function Space
