Hyponormal operator
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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 ([math]\displaystyle{ 0 \lt p \le 1 }[/math]) if:
- [math]\displaystyle{ (T^*T)^p \ge (TT^*)^p }[/math]
(That is to say, [math]\displaystyle{ (T^*T)^p - (TT^*)^p }[/math] is a positive operator.) If [math]\displaystyle{ p = 1 }[/math], then T is called a hyponormal operator. If [math]\displaystyle{ p = 1/2 }[/math], then T is called a semi-hyponormal operator. Moreover, T is said to be log-hyponormal if it is invertible and
- [math]\displaystyle{ \log (T^*T) \ge \log (TT^*). }[/math]
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 (1997). "A Note on p-Hyponormal Operators". Proceedings of the American Mathematical Society 125 (12): 3617–3624. doi:10.1090/S0002-9939-97-04004-5.
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