Aluthge transform explained

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| ^{(12)/()} U|T| ^{(12)/()} .

More generally, for any real number

λ \in [ 0 , 1 ] , the λ -Aluthge transformation is defined as \Delta ^{λ} _{λ (T): = |T|} U|T| ^{1-λ} \in B(H).

Example

For vectors

x,y \in H , let x ⊗ y denote the operator defined as \forall z\in H x ⊗ y(z) = \langle z,y\rangle x.

An elementary calculation^[2] shows that if

y\ne 0 , then \Delta _{λ(x ⊗} y) = \Delta(x ⊗ y) = (\langle x,y\rangle)/(\lVert y \rVert ^{2}) y ⊗ 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.

Notes and References

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