مروری بر رده‌های عملگرهای ترکیبی

در این مقاله نخست به معرفی عملگر امید شرطی پرداخته، سپس رده‌های کلاسیک را برای عملگرهای ترکیبی و ترکیبی وزن‌دار مرور می‌کنیم. رده‌های زیادی از عملگرها روی فضای هیلبرت وجود دارند، به‌طوری‌که ضعیف‌تر از رده عملگرهای هیپونرمال هستند، مانند عملگرهای هیپونرمال، شبه‌هیپونرمال، پارانرمال، نرمالوئید و غیره، در این مقاله از دیدگاه نظریه اندازه، عملگرهای از نوع ترکیبی، ترکیبی وزن‌دار، الحاقی عملگرهای ترکیبی وزن‌دار و تبدیلات آلوثگ تعمیم‌یافته وابسته به آنها را روی فضای در نظر گرفته و شرایط لازم و کافی برای تعلق این نوع عملگرها به هرکدام از رده‌های بالا بیان می‌شود. هم‌چنین زیرنرمال بودن عملگرهای ترکیبی و ترکیبی وزن‌دار نیز بررسی می‌شود. در پایان با ارائه مثال‌هایی متنوع، نشان می‌دهیم که عملگرها این رده‌ها را تفکیک می‌کنند.

A Review on Classes of Composition Operators

IntroductionIn 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 methodsLet be a complete finite measure space and be a complete finite measure space where is a subalgebra of . For any nonnegative measurable functions as well as for any , by the RadonNikodym 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 nonsingular transformation again by the RadonNikodym theorem, there exists a nonnegative unique function such that The function is called RadonNikodym derivative of with respect . These are two most useful tools which play important roles in this review.For a nonnegative finitevalued measurable function and a nonsingular 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 discussionIn 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 RadonNikodym 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.ConclusionAccording 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