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2013-10-18T16:26:19Z

Reverted 1 edit by 12.69.178.19 (talk) to last revision by Addbot. (TW)

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In [[mathematics]], especially [[operator theory]], a '''hyponormal operator''' is a generalization of a [[normal operator]]. In general, a bounded [[linear operator]] ''T'' on a complex [[Hilbert space]] ''H'' is said to be '''''p''-hyponormal''' (<math>0 < p \le 1</math>) if:
:<math>(T^*T)^p \ge (TT^*)^p</math>
(That is to say, <math>(T^*T)^p - (TT^*)^p</math> is a positive operator.) If <math>p = 1</math>, then ''T'' is called a hyponormal operator. If <math>p = 1/2</math>, then ''T'' is called a semi-hyponormal operator. Moreoever, ''T'' is said to be '''log-hyponormal''' if it is invertible and
:<math>\log (T^*T) \ge \log (TT^*).</math>
An invertible ''p''-hyponormal operator is log-hyponormal. On the other hand, not every log-hyponormal is ''p''-hyponormal.

The class of semi-hyponormal operators was introduced by Xia, and the class of p-hyponormal operators was studied by Aluthge, who used what is today called the [[Aluthge transformation]].

Every [[subnormal operator]] (in particular, a normal operator) is hyponormal, and every hyponormal operator is a [[paranormal operator|paranormal]] [[convexoid operator]]. Not every paranormal operator is, however, hyponormal.


== See also ==
*[[Putnam’s inequality]]

== References ==
*http://www.jstor.org/pss/2162263

[[Category:Operator theory]]

{{mathanalysis-stub}}

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en>Konveyor Belt: Reverted 1 edit by 12.69.178.19 (talk) to last revision by Addbot. (TW)