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:
 formula_1
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.
Every hyponormal operator (in particular, a subnormal operator, a quasinormal operator and a normal operator) is paranormal. If "T" is a paranormal, then "T""n" is paranormal. On the other hand, Halmos gave an example of a hyponormal operator "T" such that "T"2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.
A compact paranormal operator is normal.
References.
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