Aluthge transform
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.
Definition
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
For vectors, let denote the operator defined asAn elementary calculation shows that if, then