Paranormal operator

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:

‖ T 2 x ‖ ≥ ‖ T x ‖ 2 {\displaystyle \|T^{2}x\|\geq \|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

[edit]

1. ^ Istrăţescu, V. (1967). "On some hyponormal operators". Pacific Journal of Mathematics. 22 (3): 413–417\. doi:10.2140/pjm.1967.22.413. MR 0213893.
2. ^ a b Furuta, Takayuki (1967). "On the class of paranormal operators". Proceedings of the Japan Academy. 43: 594–598\. MR 0221302.
3. ^ Halmos, Paul Richard (1982). A Hilbert Space Problem Book. Encyclopedia of Mathematics and its Applications. Vol. 17 (2nd ed.). Springer-Verlag, New York-Berlin. ISBN 0-387-90685-1. MR 0675952.
4. ^ Furuta, Takayuki (1971). "Certain convexoid operators". Proceedings of the Japan Academy. 47: 888–893\. doi:10.2183/pjab1945.47.SupplementI_888. MR 0313864.

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