// Workers AI · dad joke modeWhat did the Aluthge transform say? I'm transforming.

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