Hyponormal operator

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 (${\displaystyle 0<p\leq 1}$) if:

${\displaystyle (T^{*}T)^{p}\geq (TT^{*})^{p}}$

(That is to say, ${\displaystyle (T^{*}T)^{p}-(TT^{*})^{p}}$ is a positive operator.) If ${\displaystyle p=1}$, then T is called a hyponormal operator. If ${\displaystyle p=1/2}$, then T is called a semi-hyponormal operator. Moreover, T is said to be log-hyponormal if it is invertible and

${\displaystyle \log(T^{*}T)\geq \log(TT^{*}).}$

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.