Aluthge transform

Last updated June 07, 2021

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.^[1]

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

Notes

↑ Aluthge, Ariyadasa (1990). "On p-hyponormal operators for 0 < p < 1". Integral Equations Operator Theory. 13 (3): 307–315. doi:10.1007/bf01199886.
↑ Chabbabi, Fadil; Mbekhta, Mostafa (June 2017). "Jordan product maps commuting with the λ-Aluthge transform". Journal of Mathematical Analysis and Applications. 450 (1): 293–313. doi: 10.1016/j.jmaa.2017.01.036 .

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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

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[1] Aluthge, Ariyadasa (1990). "On p-hyponormal operators for 0 < p < 1". Integral Equations Operator Theory. 13 (3): 307–315. doi:10.1007/bf01199886.

[2] Chabbabi, Fadil; Mbekhta, Mostafa (June 2017). "Jordan product maps commuting with the λ-Aluthge transform". Journal of Mathematical Analysis and Applications. 450 (1): 293–313. doi: 10.1016/j.jmaa.2017.01.036 .

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