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.
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
For vectors , let denote the operator defined as
An elementary calculation shows that if , then