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:

$\Delta (T)=|T|^{\frac {1}{2}}U|T|^{\frac {1}{2}}.$

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 $y\neq 0$, then $\Delta _{\lambda }(x\otimes y)=\Delta (x\otimes y)={\frac {\langle x,y\rangle }{\lVert y\rVert ^{2}}}y\otimes y.$

References

• Antezana, Jorge; Pujals, Enrique R.; Stojanoff, Demetrio (2008). "Iterated Aluthge transforms: a brief survey". Revista de la Unión Matemática Argentina. 49: 29–41.

External links

• Ariyadasa Aluthge at the Mathematics Genealogy Project