The concept of partial isometry can be defined in other equivalent ways.

V can be extended to a partial isometry acting on all of .

The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues.

The union of these maps defines a partial isometry whose domain resp.

A partially defined isometric operator with closed domain is called a partial isometry.

A partial isometry V has a unitary extension if and only if the deficiency indices are identical.

The Toeplitz structure of A means that a "truncated" shift is a partial isometry on .

Any unitary operator on H is a partial isometry with initial and final subspaces being all of H.

In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry V.

When B and B are not assumed to be minimal, the same calculation shows that above claim holds verbatim with U being a partial isometry.

In general, a partial isometry may not be extendable to a unitary operator and therefore a quasinormal operator need not be normal.

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

ETF has polar decomposition UH for some partial isometry U and positive operator H in M.

The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator.

A pair of projections one of which is the initial projection of a partial isometry and the other a final projection of the same isometry are said to be equivalent.

The matrix A is the partial isometry that vanishes on the orthogonal complement of U and A is the isometry that embeds U into the underlying vector space.

In functional analysis a partial isometry is a linear map W between Hilbert spaces H and K such that the restriction of W to the orthogonal complement of its kernel is an isometry.

Two subspaces belonging to M are called (Murray-von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if M "knows" that the subspaces are isomorphic).