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Hyponormal operator

Generalized normal operator

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Source updated July 2, 2024

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 ( $0<p\leq 1$ ) if:

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

(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

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

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.

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