This page is about the polar decomposition of bounded operators on Hilbert spaces. Any complex number has a representation as with being the absolute value of and the complex number of norm being the modulus, or the complex sign, of . The polar decomposition of a bounded operator is a generalization of this representation.
Let be a Hilbert space and be closed linear subspaces.
An unitary isomorphism
is called a partial isometry with initial space and final space or range
Let be a bounded operator on
The positive operator
is called the modulus of T.
For every bounded operator on there exists a unique partial isometry such that
U has initial space and range
This is true however for every von Neumann algebra.
Most textbooks about operators on Hilbert spaces mention the polar decomposition, for example it can be found in the beginning of
Kadison, Ringrose: Fundamentals of the Theory of Operator Algebras , volume 2, Advanced Theory
wikipedia polar decomposition