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

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

(That is to say, $${\displaystyle (T^{*}T)^{p}-(TT^{*})^{p}}$$ is a positive operator.) If 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

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

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.

Putnam’s inequality

References

http://www.jstor.org/pss/2162263