Paranormal operator explained

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]

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]

Notes and References

1. Istrăţescu . V. . Pacific Journal of Mathematics . 213893 . 413–417 . On some hyponormal operators . 22 . 1967.
2. Furuta . Takayuki . Proceedings of the Japan Academy . 221302 . 594–598 . On the class of paranormal operators . 43 . 1967.
3. Book: Halmos, Paul Richard . 2nd . 0-387-90685-1 . 675952 . Springer-Verlag, New York-Berlin . Encyclopedia of Mathematics and its Applications . A Hilbert Space Problem Book . 17 . 1982.
4. Furuta . Takayuki . Proceedings of the Japan Academy . 313864 . 888–893 . Certain convexoid operators . 47 . 1971.