Hyponormal operator

{{Short description|Generalized normal operator}}

{{Onesource|article|date=February 2022}}{{Nofootnotes|article|date=February 2022}}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 () if:

:$(T^*T)^p\; \backslash ge\; (TT^*)^p$

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

:$\backslash log\; (T^*T)\; \backslash ge\; \backslash log\; (TT^*).$

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.