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

be a Hilbert space and let

B(H)

be the algebra of linear operators from

. By the polar decomposition theorem, there exists a unique partial isometry

such that

T=U|T|

and

\ker(U)\supset\ker(T)

, where

|T|

is the square root of the operator

T^*T

. If

T\inB(H)

and

T=U|T|

is its polar decomposition, the Aluthge transform of

is the operator

\Delta(T)

defined as:

	12

\Delta(T)=|T|

	12

U|T|

More generally, for any real number

λ\in[0,1]

, the

-Aluthge transformation is defined as

	λ
\Delta
	λ(T):=\|T\|

U|T|^1-λ\inB(H).

Example

For vectors

x,y\inH

, let

x ⊗ y

denote the operator defined as

\forallz\inH x ⊗ y(z)=\langlez,y\ranglex.

An elementary calculation^[2] shows that if

y\ne0

, then

\Delta_{λ(x ⊗}y)=\Delta(x ⊗ y)=

	\langlex,y\rangle
	\lVerty\rVert²

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

Aluthge. Ariyadasa. 1990. On p-hyponormal operators for 0 < p < 1. Integral Equations Operator Theory. 13. 3. 307–315. 10.1007/bf01199886.
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.