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 formula_1 be a Hilbert space and let formula_2 be the algebra of linear operators from formula_1 to formula_1. By the polar decomposition theorem, there exists a unique partial isometry formula_5 such that formula_6 and formula_7, where formula_8 is the square root of the operator formula_9. If formula_10 and formula_11 is its polar decomposition, the Aluthge transform of formula_12 is the operator formula_13 defined as:
More generally, for any real number formula_15, the formula_16-Aluthge transformation is defined as 
Example.
For vectors formula_18, let formula_19 denote the operator defined as 
An elementary calculation shows that if formula_21, then formula_22