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Optimal Extensions for pth Power Factorable Operators
Let X ( μ ) be a function space related to a measure space ( Ω , Σ , μ ) with χ Ω ∈ X ( μ ) and let T : X ( μ ) → E be a Banach space-valued operator. It is known that if T is p th power factorable then the largest function space to which T can be extended preserving p th power factorability is give...
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| Published in: | Mediterranean journal of mathematics 2016-12, Vol.13 (6), p.4281-4303 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Let
X
(
μ
)
be a function space related to a measure space
(
Ω
,
Σ
,
μ
)
with
χ
Ω
∈
X
(
μ
)
and let
T
:
X
(
μ
)
→
E
be a Banach space-valued operator. It is known that if
T
is
p
th power factorable then the largest function space to which
T
can be extended preserving
p
th power factorability is given by the space
L
p
(
m
T
) of
p
-integrable functions with respect to
m
T
, where
m
T
:
Σ
→
E
is the vector measure associated to
T
via
m
T
(
A
)
=
T
(
χ
A
)
. In this paper, we extend this result by removing the restriction
χ
Ω
∈
X
(
μ
)
. In this general case, by considering
m
T
defined on a certain
δ
-ring, we show that the optimal domain for
T
is the space
L
p
(
m
T
)
∩
L
1
(
m
T
)
. We apply the obtained results to the particular case when
T
is a map between sequence spaces defined by an infinite matrix. |
|---|---|
| ISSN: | 1660-5446 1660-5454 |
| DOI: | 10.1007/s00009-016-0745-1 |