In mathematics and more precisely in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, the Aluthge transformation is an operation defined on the set of

bounded operators In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

of a Hilbert space. It was introduced by Ariyadasa Aluthge to study p-hyponormal linear operators.

Definition

Let

H

be a Hilbert space and let

B(H)

be the algebra of linear operators from

H

H

. By the

polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...

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, ^U, T, ^.

More generally, for any real number

\lambda\in,1 /math>, the \lambda -Aluthge transformation is defined as 
: \Delta_\lambda(T):=, T, ^U, T, ^\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 shows that if

y\ne0

, then

\Delta_\lambda(x\otimes y)=\Delta(x\otimes y)=\frac y\otimes y.

Notes

References

External links

* {{MathGenealogy, id=59270, 59270, title=Ariyadasa Aluthge, Ariyadasa Aluthge Bilinear forms Matrices Topology