Paranormal operator

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:

\|T^2x\| \ge \|Tx\|^2 \,

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

  1. ↑ V. Istratescu. On some hyponormal operators
  2. 1 2 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
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