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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:
 
: <math>\|T^2x\| \ge \|Tx\|^2 \,</math>
 
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.<ref>V. Istratescu. ''[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102992095 On some hyponormal operators]''</ref><ref name="Furuta" />
 
Every [[hyponormal operator]] (in particular, a [[subnormal operator]], a [[quasinormal operator]] and a normal operator) is paranormal. If ''T'' is a paranormal, then ''T''<sup>''n''</sup> is paranormal.<ref name="Furuta">Furuta, Takayuki. ''[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pja/1195521514 On the Class of Paranormal Operators]''</ref> On the other hand, [[Paul Halmos|Halmos]] gave an example of a hyponormal operator ''T'' such that ''T''<sup>2</sup> isn't hyponormal. Consequently, not every paranormal operator is hyponormal.<ref>P.R.Halmos, ''A Hilbert Space Problem Book'' 2nd edition, Springer-Verlag, New York, 1982.</ref>
 
A [[Compact operator|compact]] paranormal operator is normal.<ref>Furuta, Takayuki. [http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=pjab1945&cdvol=47&noissue=SupplementI&startpage=888&lang=en&from=jnlabstract Certain Convexoid Operators]</ref>
 
== References ==
{{reflist}}
 
[[Category:Operator theory]]
 
{{mathanalysis-stub}}

Revision as of 20:10, 14 December 2013

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:

T2xTx2

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.[1][2]

Every hyponormal operator (in particular, a subnormal operator, a quasinormal operator and a normal operator) is paranormal. If T is a paranormal, then Tn is paranormal.[2] On the other hand, Halmos gave an example of a hyponormal operator T such that T2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.[3]

A compact paranormal operator is normal.[4]

References

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Template:Mathanalysis-stub

  1. V. Istratescu. On some hyponormal operators
  2. 2.0 2.1 Furuta, Takayuki. On the Class of Paranormal Operators
  3. P.R.Halmos, A Hilbert Space Problem Book 2nd edition, Springer-Verlag, New York, 1982.
  4. Furuta, Takayuki. Certain Convexoid Operators