Mathematical Researches

fa مروری بر رده‌های عملگرهای ترکیبی A Review on Classes of Composition Operators جبر alg مقاله مستقل Original Manuscript 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 <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > - هیپونرمال، <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > - شبه‌هیپونرمال، <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > - 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 <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" >  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.   کد موضوع‌بندی ریاضی (2010): 37B47، 47B20 Introduction In 1976, A. Lambert characterized subnormal weighted shifts. Then he studied hyponormal weighted composition operators on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="35" > 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  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="16" > is normal if and only if   <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="62" >  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 <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="51" > be a complete <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="7" > -finite measure space and  <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="54" > be a complete <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="7" > -finite measure space where <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="16" > is a subalgebra  of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="9" >. For any non-negative <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="9" >-measurable functions <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="8" > as well as for any <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="66" >, by the Radon-Nikodym theorem, there exists a unique <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="13" >-measurable function<img alt="" height="22" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="44" > such that <img alt="" height="27" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="143" > for all <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="46" > <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image030.gif" width="4" >As an operator,  <img alt="" height="22" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image032.gif" width="134" > is a contractive orthogonal projection which is called the conditional expectation operator with respect <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image034.gif" width="16" >  For a non-singular transformation <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image036.gif" width="58" > again by the Radon-Nikodym theorem, there exists a non-negative  unique function <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image038.gif" width="61" > such that <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="174" > The function <img alt="" height="31" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image042.gif" width="68" >  is called Radon-Nikodym derivative of  <img alt="" height="21" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image044.gif" width="48" > with respect <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image046.gif" width="8" >. These are two most useful tools which play important roles in this review. For a non-negative finite-valued <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="9" > - measurable function <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image048.gif" width="9" > and a non-singular transformation <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image036.gif" width="58" > the weighted composition operator <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image050.gif" width="15" > on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="35" > induced by <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image052.gif" width="9" > and <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image054.gif" width="15" >is given by <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="139" >, where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="16" > is called the composition operator on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="35" >. <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image058.gif" width="18" >is bounded on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image060.gif" width="37" > for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image062.gif" width="65" >  if and only if <img alt="" height="22" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image064.gif" width="186" > Results and discussion In this paper, we review some known classes of composition operators, weighted composition operators, their adjoints and Aluthge transformations on <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="35" > such as normal, subnormal, normaloid, hyponormal, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image066.gif" width="8" >-hyponormal, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image066.gif" width="8" >-quasihyponormal, <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image066.gif" width="8" >-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 <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image068.gif" width="7" > and the conditional expectation operator <img alt="" height="22" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image070.gif" width="20" > 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.<a href="./files/site1/files/62/10Abstract.pdf">./files/site1/files/62/10Abstract.pdf</a>   عملگرهای ترکیبی, امید شرطی, نرمال, زیرنرمال, هیپونرمال, ضعیف هیپونرمال. Composition operators, Conditional expectation, Normal, Subnormal, Hyponormal, Weakly hyponormal. 243 260 http://mmr.khu.ac.ir/browse.php?a_code=A-10-157-1&slc_lang=fa&sid=1 Mohammadreza Azimi محمدرضا عظیمی mhr.azimi@maragheh.ac.ir 10031947532846003743 10031947532846003743 Yes University of Maragheh دانشگاه مراغه، دانشکدۀ علوم پایه، گروه ریاضی