Basic properties of multiplication and composition operators between distinct Orlicz spaces

• Published in: Revista matemática complutense Vol. 30; no. 2; pp. 335 - 367
• Main Authors: Chawziuk, T., Estaremi, Y., Hudzik, H., Maghsoudi, S., Rahmani, I.
• Format: Journal Article
• Language: English
• Published: Milan Springer Milan 01-05-2017; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Multiplication; Operators (mathematics); Orlicz space; Topology; Measurable transformation; Composition operator; Lebesgue space; 47B33; Finite-rank operators; Musielak–Orlicz space; 47B38; Multiplication operator; 46E30; Closed-range operators; Orlicz space; Radon–Nikodym derivative; Continuous operators
• ISSN: 1139-1138; 1988-2807
• DOI: 10.1007/s13163-016-0214-1

First, we present some simple (and easily verifiable) necessary conditions and sufficient conditions for boundedness of the multiplication operator
M
u
and composition operator
C
T
acting from Orlicz space
L
Φ
1
(
Ω
)
into Orlicz space
L
Φ
2
(
Ω
)
over arbitrary complete,
σ
-finite measure space
(
Ω
,
Σ
,
μ
)
. Next, we investigate the problem of conditions on the generating Young functions, the function
u
, and/or the function
h
=
d
(
μ
∘
T
-
1
)
/
d
μ
, under which the operators
M
u
and
C
T
are of closed range or finite rank. Finally, we give necessary and sufficient conditions for boundedness of the operators
M
u
and
C
T
in terms of techniques developed within the theory of Musielak–Orlicz spaces.