In mathematics, especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator T on a complex Hilbert space H is said to be p-hyponormal ( 0 < p ≤ 1 {\displaystyle 0<p\leq 1} ) if:
( T ∗ T ) p ≥ ( T T ∗ ) p {\displaystyle (T^{*}T)^{p}\geq (TT^{*})^{p}}
(That is to say, ( T ∗ T ) p − ( T T ∗ ) p {\displaystyle (T^{*}T)^{p}-(TT^{*})^{p}} is a positive operator.) If p = 1 {\displaystyle p=1} , then T is called a hyponormal operator. If p = 1 / 2 {\displaystyle p=1/2} , then T is called a semi-hyponormal operator. Moreover, T is said to be log-hyponormal if it is invertible and
log ( T ∗ T ) ≥ log ( T T ∗ ) . {\displaystyle \log(T^{*}T)\geq \log(TT^{*}).}
An invertible p-hyponormal operator is log-hyponormal. On the other hand, not every log-hyponormal is p-hyponormal. The class of semi-hyponormal operators was introduced by Xia, and the class of p-hyponormal operators was studied by Aluthge, who used what is today called the Aluthge transformation. Every subnormal operator (in particular, a normal operator) is hyponormal, and every hyponormal operator is a paranormal convexoid operator. Not every paranormal operator is, however, hyponormal.
References Huruya, Tadasi (1997). "A Note on p-Hyponormal Operators". Proceedings of the American Mathematical Society. 125 (12): 3617–3624. doi:10.1090/S0002-9939-97-04004-5. JSTOR 2162263.