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Aluthge transform

From Wikipedia, the free encyclopedia

In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by Ariyadasa Aluthge to study p-hyponormal linear operators.[1]

Definition

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Let be a Hilbert space and let be the algebra of linear operators from to . By the polar decomposition theorem, there exists a unique partial isometry such that and , where is the square root of the operator . If and is its polar decomposition, the Aluthge transform of is the operator defined as:

More generally, for any real number , the -Aluthge transformation is defined as

Example

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For vectors , let denote the operator defined as

An elementary calculation[2] shows that if , then

Notes

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  1. ^ Aluthge, Ariyadasa (1990). "On p-hyponormal operators for 0 < p < 1". Integral Equations Operator Theory. 13 (3): 307–315. doi:10.1007/bf01199886.
  2. ^ Chabbabi, Fadil; Mbekhta, Mostafa (June 2017). "Jordan product maps commuting with the λ-Aluthge transform". Journal of Mathematical Analysis and Applications. 450 (1): 293–313. doi:10.1016/j.jmaa.2017.01.036.

References

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