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

[math]\displaystyle{ \Delta(T)=|T|^{\frac12}U|T|^{\frac12}. }[/math]

More generally, for any real number [math]\displaystyle{ \lambda\in [0,1] }[/math], the [math]\displaystyle{ \lambda }[/math]-Aluthge transformation is defined as

[math]\displaystyle{ \Delta_\lambda(T):=|T|^{\lambda}U|T|^{1-\lambda}\in B(H). }[/math]

Example

For vectors [math]\displaystyle{ x,y \in H }[/math], let [math]\displaystyle{ x\otimes y }[/math] denote the operator defined as

[math]\displaystyle{ \forall z\in H\quad x\otimes y(z)=\langle z,y\rangle x. }[/math]

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

References

• Antezana, Jorge; Pujals, Enrique R.; Stojanoff, Demetrio (2008). "Iterated Aluthge transforms: a brief survey". Revista de la Unión Matemática Argentina 49: 29–41. http://www.scielo.org.ar/scielo.php?script=sci_arttext&pid=S0041-69322008000100004. 

External links

• Ariyadasa Aluthge at the Mathematics Genealogy Project