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 $$H$$ be a Hilbert space and let  $$B(H)$$ be the algebra of linear operators from $$H$$ to $$H$$. By the polar decomposition theorem, there exists a unique partial isometry $$U$$ such that  $$T=U|T|$$ and  $$\ker(U)\supset\ker(T)$$, where  $$|T|$$ is the square root of the operator  $$ T^*T$$. If $$T\in B(H)$$ and $$ T=U|T|$$ is its polar decomposition, the Aluthge transform of  $$T$$ is the operator $$\Delta(T)$$ defined as:
 * $$\Delta(T)=|T|^{\frac12}U|T|^{\frac12}.$$

More generally, for any real number $$\lambda\in [0,1]$$, the $$\lambda$$-Aluthge transformation is defined as
 * $$\Delta_\lambda(T):=|T|^{\lambda}U|T|^{1-\lambda}\in B(H).$$

Example
For vectors $$x,y \in H$$, let $$x\otimes y$$ denote the operator defined as
 * $$\forall z\in H\quad x\otimes y(z)=\langle z,y\rangle x.$$

An elementary calculation shows that if $$y\ne0$$, then $$\Delta_\lambda(x\otimes y)=\Delta(x\otimes y)=\frac{\langle x,y\rangle}{\lVert y \rVert^2} y\otimes y.$$