A Review on Classes of Composition Operators

Author
Mohammadreza Azimi
University of Maragheh
10.52547/mmr.6.2.243
Abstract
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

Keywords
Composition operators
Conditional expectation
  Normal
Subnormal
Hyponormal
  Weakly  hyponormal
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Mathematical Research
Volume 6, Issue 2 - Serial Number 13
Summer 2020
Pages 243-260