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:

${\displaystyle \|T^{2}x\|\geq \|Tx\|^{2}}$

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]