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Aluthge transform

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Source updated April 26, 2026

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.

01Definition

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

$\Delta _{\lambda }(T):=|T|^{\lambda }U|T|^{1-\lambda }\in B(H).$

02Example

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

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This article is adapted from the Wikipedia article “Aluthge transform”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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