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.<ref>{{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}}</ref>
== Definition == Let <math>H</math> be a Hilbert space and let <math>B(H)</math> be the algebra of linear operators from <math>H</math> to <math>H</math>. By the polar decomposition theorem, there exists a unique partial isometry <math>U</math> such that <math>T=U|T|</math> and <math>\ker(U)\supset\ker(T)</math>, where <math>|T|</math> is the square root of the operator <math> T^*T</math>. If <math>T\in B(H)</math> and <math> T=U|T|</math> is its polar decomposition, the Aluthge transform of <math>T</math> is the operator <math>\Delta(T)</math> defined as: : <math>\Delta(T)=|T|^{\frac12}U|T|^{\frac12}.</math>
More generally, for any real number <math>\lambda\in [0,1]</math>, the <math>\lambda</math>-Aluthge transformation is defined as : <math>\Delta_\lambda(T):=|T|^{\lambda}U|T|^{1-\lambda}\in B(H).</math>
== Example == For vectors <math>x,y \in H</math>, let <math>x\otimes y</math> denote the operator defined as : <math>\forall z\in H\quad x\otimes y(z)=\langle z,y\rangle x.</math>
An elementary calculation<ref>{{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= }}</ref> shows that if <math>y\ne0</math>, then <math>\Delta_\lambda(x\otimes y)=\Delta(x\otimes y)=\frac{\langle x,y\rangle}{\lVert y \rVert^2} y\otimes y.</math>
== Notes == {{Reflist}}
== 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}}
Category:Bilinear forms Category:Matrices (mathematics) Category:Topology