Hyponormal operator explained

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 (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 andlog (T ^{*T)} \ge 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.

References

• Huruya . Tadasi . A Note on p-Hyponormal Operators . Proceedings of the American Mathematical Society . 1997 . 125 . 12 . 3617–3624 . 10.1090/S0002-9939-97-04004-5 . 2162263 . free .