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:

‖T2x‖≥‖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. ↑ Istrăţescu, V. (1967). "On some hyponormal operators". Pacific Journal of Mathematics 22 (3): 413–417. doi:10.2140/pjm.1967.22.413. https://projecteuclid.org/euclid.pjm/1102992095.
2. ↑ 2.0 2.1 Furuta, Takayuki (1967). "On the class of paranormal operators". Proceedings of the Japan Academy 43: 594–598. https://projecteuclid.org/euclid.pja/1195521514.
3. ↑ Halmos, Paul Richard (1982). A Hilbert Space Problem Book. Encyclopedia of Mathematics and its Applications. 17 (2nd ed.). Springer-Verlag, New York-Berlin. ISBN 0-387-90685-1.
4. ↑ Furuta, Takayuki (1971). "Certain convexoid operators". Proceedings of the Japan Academy 47: 888–893. doi:10.2183/pjab1945.47.SupplementI_888. https://projecteuclid.org/euclid.pja/1195526397.

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