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.{{Cite journal|last=Aluthge|first=Ariyadasa|date=1990|title=On p-hyponormal operators for 0 < p < 1|journal=Integral Equations Operator Theory|volume=13|issue=3|pages=307–315|doi=10.1007/bf01199886}}

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:

: $\Delta(T)=|T|^{\frac12}U|T|^{\frac12}.$

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{{cite journal |last1=Chabbabi |first1=Fadil |last2=Mbekhta |first2=Mostafa |title=Jordan product maps commuting with the λ-Aluthge transform |journal=Journal of Mathematical Analysis and Applications |date=June 2017 |volume=450 |issue=1 |pages=293–313 |doi=10.1016/j.jmaa.2017.01.036|doi-access= }} shows that if $y\ne0$ , then $\Delta_\lambda(x\otimes y)=\Delta(x\otimes y)=\frac{\langle x,y\rangle}{\lVert y \rVert^2} y\otimes y.$

Notes

References

{{Cite journal|last=Antezana|first=Jorge|last2=Pujals|first2=Enrique R.|last3=Stojanoff|first3=Demetrio|date=2008|title=Iterated Aluthge transforms: a brief survey|url=http://www.scielo.org.ar/scielo.php?script=sci_arttext&pid=S0041-69322008000100004|journal=Revista de la Unión Matemática Argentina|volume=49|pages=29–41}}

External links

{{MathGenealogy|id=59270|59270|title=Ariyadasa Aluthge|Ariyadasa Aluthge}}

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