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 ...

, 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 sp ...

of a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

. It was introduced by Ariyadasa Aluthge to study p-hyponormal

linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

Definition

Let

H

be a

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 ...

theorem, there exists a unique

partial isometry In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace and its range is cal ...

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