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

$(T^{*}T{)}^p\ge (T{T}^{*}{)}^p$

(That is to say, $(T^{*}T{)}^p-(T{T}^{*}{)}^p$ 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. Moreoever, T is said to be log-hyponormal if it is invertible and

$\log (T^{*}T)\ge \log (T{T}^{*}).$

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.

See also

• Putnam’s inequality

References

• http://www.jstor.org/pss/2162263

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