Paranormal operator

Short description: Operator type

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]\displaystyle{ \|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.^[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

↑ V. Istratescu. On some hyponormal operators
↑ ^2.0 ^2.1 Furuta, Takayuki. On the Class of Paranormal Operators
↑ P.R.Halmos, A Hilbert Space Problem Book 2nd edition, Springer-Verlag, New York, 1982.
↑ Furuta, Takayuki. Certain Convexoid Operators^{[yes|permanent dead link|dead link}}]}

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[1] V. Istratescu. On some hyponormal operators

[Furuta-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^{[yes|permanent dead link|dead link}}]}

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