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
be a
Hilbert space and let
be the algebra of linear operators from
to
. 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 such that
and
, where
is the
square root of the operator . If
and
is its polar decomposition, the Aluthge transform of
is the operator
defined as:
:
More generally, for any real number