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In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. More precisely, a bounded linear operator ''T'' on a complex Hilbert space ''H'' is said to be paranormal if: : for every unit vector ''x'' in ''H''. The class of paranormal operators was introduced by V. Istratescu in 1960s, though the term "paranormal" is probably due to Furuta.〔V. Istratescu. ''(On some hyponormal operators )''〕〔 Every hyponormal operator (in particular, a subnormal operator, a quasinormal operator and a normal operator) is paranormal. If ''T'' is a paranormal, then ''T''''n'' is paranormal.〔Furuta, Takayuki. ''(On the Class of Paranormal Operators )''〕 On the other hand, Halmos gave an example of a hyponormal operator ''T'' such that ''T''2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.〔P.R.Halmos, ''A Hilbert Space Problem Book'' 2nd edition, Springer-Verlag, New York, 1982.〕 A compact paranormal operator is normal.〔Furuta, Takayuki. (Certain Convexoid Operators )〕 == References == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Paranormal operator」の詳細全文を読む スポンサード リンク
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