American Journal of Mathematical Analysis, 2015, Vol. 3, No. 1, 21-25
Available online at http://pubs.sciepub.com/ajma/3/1/5
© Science and Education Publishing
DOI:10.12691/ajma-3-1-5
Composition Operators on Lorentz-Karamata-Bochner
Spaces
ANURADHA GUPTA
1
, NEHA BHATIA
2,*
1
Department of Mathematics, Delhi College of Arts and Commerce, University of Delhi, Delhi-110023, India
2
Department of Mathematics, University of Delhi, Delhi -110007, India
*
Corresponding author: nehaphd@yahoo.com
Received January 10, 2015; Revised February 10, 2015; Accepted February 27, 2015
Abstract In this paper, study of the composition operators on Lorentz-Karamata-Bochner spaces and
characterization of the properties like boundedness, closedness and essential range of these operators on the space
has been made.
Keywords: slowly varying function, Lorentz-Karamata-Bochner spaces, composition operators, closed range,
essential range
Cite This Article: ANURADHA GUPTA, and NEHA BHATIA, “Composition Operators on Lorentz-
Karamata-Bochner Spaces.” American Journal of Mathematical Analysis, vol. 3, no. 1 (2015): 21-25. doi:
10.12691/ajma-3-1-5.
1. Introduction
Let ( ) , , µ Ω be a σ-finite measure space. A
measurable transformation T is said to be non-singular if
( )
( )
1
0 T A µ
−
= whenever ( ) 0 A µ = for every A ∈ .
If T is non-singular, then we say that
1
T µ
−
is
absolutely continuous with respect to µ. Hence, by Radon-
Nikodym theorem there exists a unique non-negative
essentially bounded function f
T
such that
( )
( )
1
T
A
T A fd µ µ
−
=
∫
for A ∈ .
Let f be any complex-valued measurable function. For s
≥ 0, the distribution function µ
f
of f is defined as
() ( ) { }
:| |
f
s f s µ µω ω = ∈Ω > .
The non-increasing rearrangement f
∗
of f is defined
as
() () { }
inf 0: , 0.
f
f t s s t for all t µ
∗
= > ≤ ≥
The maximal (average) operator is given by
() ()
0
1 t
f t f s ds
t
∗∗ ∗
=
∫
.
One can refer to [4] for the properties of these functions.
Definition 1. A positive and Lebesgue measurable
function b is said to be slowly varying (s.v.) on (0, ∞) if,
for each > 0, tb(t) is equivalent to a non-decreasing
function and t–b(t) is equivalent to a non-increasing
function on (0, ∞).
Given a s.v. function b on (0, ∞), we denote by γ
b
the
positive function defined by
()
1
max ,
b
t b t
t
γ
=
For various properties of slowly varying function we
can refer to [4,10].
For 1 < p < ∞, 1 ≤ q < ∞ and for a measurable function
f on Ω, define
() ()
( )
() ()
1 1
,;
; 0,
1
1 1
0
|| ||
p q
pq b
q
q
q
p q
b
f t t f t
t t f t dt
α
γ
γ
−
∗∗
∞
−
∞
∗∗
=
=
∫
The Lorentz-Karamata space
,; pqb
L introduced in [4] is
the set of all measurable functions f on Ω such that
,; pqb
f <∞ .
Let : f X Ω→ be a strongly measurable function on a
Banach space X. Define a function f as
( ) ( ) f w f w =
for all
ω ∈Ω . Then the Lorentz-Karamata-Bochner space
( )
,;
,
pqb
L X Ω is a rearrangement invariant-Bochner space
for ( ) , 0, pq ∈ ∞ where the norm is given as
() ()
( )
1 1
,;
; 0,
|
p q
b
pqb
q
f t t f t γ
−
∗∗
∞
=