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 called the final subspace.
Partial isometries appear in the polar decomposition.
General.
The concept of partial isometry can be defined in other equivalent ways. If "U" is an isometric map defined on a closed subset "H"1 of a Hilbert space "H" then we can define an extension "W" of "U" to all of "H" by the condition that "W" be zero on the orthogonal complement of "H"1. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map.
Partial isometries (and projections) can be defined in the more abstract setting of a semigroup with involution; the definition coincides with the one herein.
Operator Algebras.
For operator algebras one introduces the initial and final subspaces:
C*-Algebras.
For C*-algebras one has the chain of equivalences due to the C*-property:
So one defines partial isometries by either of the above and declares the initial resp. final projection to be W*W resp. WW*.
A pair of projections are partitioned by the equivalence relation:
It plays an important role in K-theory for C*-algebras and in the Murray-von Neumann theory of projections in a von Neumann algebra.
Special Classes.
Projections.
Any orthogonal projection is one with common initial and final subspace:
Embeddings.
Any isometric embedding is one with full initial subspace:
Unitaries.
Any unitary operator is one with full initial and final subspace:
"(Apart from these there are far more partial isometries.)"
Examples.
Nilpotents.
On the two-dimensional complex Hilbert space the matrix
is a partial isometry with initial subspace
and final subspace
Leftshift and Rightshift.
On the square summable sequences the operators
which are related by
are partial isometries with initial subspace
and final subspace: