Introduction
In 1976, A. Lambert characterized subnormal weighted shifts. Then he studied hyponormal weighted composition operators on
in 1986 and in 1988 subnormal composition operators studied again by him. Recently, A. Lambert, et al., have published an interesting paper: Separation partial normality classes with composition operators (2005). In 1978, R. Whitley showed that a composition operator
is normal if and only if
essentially. Normal and quasinormal weighted composition operators were worked by J.T. Campbell, et al. in 1991. In 1993, J.T. Campbell, et al. worked also seminormal composition operators. Burnap C. and Jung I.B. studied composition operators with weak hyponormality in 2008.
Material and methods
Let
be a complete
-finite measure space and
be a complete
-finite measure space where
is a subalgebra of
. For any non-negative
-measurable functions
as well as for any
, by the Radon-Nikodym theorem, there exists a unique
-measurable function
such that
for all
As an operator,
is a contractive orthogonal projection which is called the
conditional expectation operator with respect
For a non-singular transformation
again by the Radon-Nikodym theorem, there exists a non-negative unique function
such that
The function
is called
Radon-Nikodym derivative of
with respect
. These are two most useful tools which play important roles in this review.
For a non-negative finite-valued
- measurable function
and a non-singular transformation
the
weighted composition operator on
induced by
and
is given by
,
where
is called the
composition operator on
.
is bounded on
for
if and only if
Results and discussion
In this paper, we review some known classes of composition operators, weighted composition operators, their adjoints and Aluthge transformations on
such as normal, subnormal, normaloid, hyponormal,
-hyponormal,
-quasihyponormal,
-paranormal, and weakly hyponormal, Furthermore, miscellaneous examples are given to illustrate that weighted composition operators lie between these classes. We discuss from the point of view of measure theory and all results depend strongly to the Radon-Nikodym derivative
and the conditional expectation operator
with their various types. Hence we study their fundamental properties in sections 1 and 2. Then, we review some results by A. Lambert, D.J. Harringston, R. Whitley, J.T. Campbell and W.E. Hornor.
Conclusion
According to the given miscellaneous examples in the final section, we can conclude that composition and weighted composition operators lie between these classes.
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Type of Study:
Original Manuscript |
Subject:
alg Received: 2017/04/22 | Revised: 2020/09/7 | Accepted: 2018/07/23 | Published: 2018/09/1 | ePublished: 2018/09/1