نرمال

In this paper, we discuss matrix theoretic characterization for weighted conditional operators by properties of conditional expectation operator in some operator classes on $L^{2}(\Sigma)$-semi-Hilbertian space such as self-adjoint, isometry and normal classes of these type operators on this space. Also, we consider the matrix representation of the Moore-Penrose inverse for these types of operators. We also gave examples to show our results.

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.

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

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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