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 $\mathrm{\Delta }(T)$ defined as:

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

More generally, for any real number $\lambda \in [0,1]$, the $\lambda$-Aluthge transformation is defined as

${\mathrm{\Delta }}_{\lambda }(T):=|T{|}^{\lambda }U|T{|}^{1-\lambda }\in B(H).$

Example

For vectors $x,y\in H$, let $x\otimes y$ denote the operator defined as

$\mathrm{\forall }z\in Hx\otimes y(z)=⟨z,y⟩x.$

An elementary calculation shows that if $y\ne 0$, then ${\mathrm{\Delta }}_{\lambda }(x\otimes y)=\mathrm{\Delta }(x\otimes y)=\frac{⟨x,y⟩}{\Vert y{\Vert }^2}y\otimes y.$

References

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External links

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