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Azimi M. A Review on Classes of Composition Operators. mmr 2020; 6 (2) :243-260
URL: http://mmr.khu.ac.ir/article-1-2618-en.html
URL: http://mmr.khu.ac.ir/article-1-2618-en.html
University of Maragheh , mhr.azimi@maragheh.ac.ir
Abstract: (2623 Views)
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../files/site1/files/62/10Abstract.pdf
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../files/site1/files/62/10Abstract.pdf
Keywords: Composition operators, Conditional expectation, Normal, Subnormal, Hyponormal, Weakly hyponormal.
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
Received: 2017/04/22 | Revised: 2020/09/7 | Accepted: 2018/07/23 | Published: 2018/09/1 | ePublished: 2018/09/1
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