Partial isometry
A partial isometry is a linear operator that preserves norms on the orthogonal complement of its kernel. Partial isometries occur naturally in Hilbert space theory, operator algebras, and the polar decomposition of bounded operators. Their defining property permits an operator to act isometrically on one closed subspace while vanishing on its orthogonal complement.
Let (H) and (K) be Hilbert spaces, and let (V:H\to K) be a bounded linear operator. The operator (V) is a partial isometry when
[ |Vx|=|x| ]
for every (x\in(\ker V)^\perp). The subspace
[ \mathcal I_V=(\ker V)^\perp ]
is the initial space of (V), while
[ \mathcal F_V=\overline{\operatorname{ran}V} ]
is its final space. The restriction (V|_{\mathcal I_V}) is an isometric isomorphism from (\mathcal I_V) onto (\mathcal F_V), and (V) is zero on (\ker V). Consequently, the range of a partial isometry is closed, so the closure in the definition of (\mathcal F_V) is redundant once the partial-isometry property has been established.
Projection characterization
The defining norm condition is equivalent to a pair of identities involving the adjoint operator. An operator (V) is a partial isometry if and only if (V^*V) is an orthogonal projection. In that case,
[ V^*V=P_{\mathcal I_V}, ]
where (P_{\mathcal I_V}) denotes the orthogonal projection onto the initial space. The corresponding product in the opposite order is also a projection:
[ VV^*=P_{\mathcal F_V}. ]
These formulas imply
[ VV^V=V \qquad\text{and}\qquad V^VV^=V^. ]
Conversely, the identity (VV^*V=V) implies that (V^V) is a projection and therefore that (V) is a partial isometry. The operator (V^) is itself a partial isometry whose initial space is (\mathcal F_V) and whose final space is (\mathcal I_V).
This characterization separates the geometric content of a partial isometry into two projections and an isometric correspondence between their ranges. The initial projection records where the operator acts nontrivially, whereas the final projection records where its values lie.
Finite-dimensional form
For a linear transformation (V:\mathbb C^m\to\mathbb C^n), partial isometry is equivalent to the condition that every singular value of (V) equals either (0) or (1). Thus there are unitary matrices (U) and (W) for which
[ V
U \begin{pmatrix} I_r & 0\ 0 & 0 \end{pmatrix} W^*, ]
with the rectangular block sizes determined by (m), (n), and the rank (r) of (V). The identity block represents the isometric action from the initial space to the final space, while the zero block represents the kernel.
This form also gives
[ V^*V
W \begin{pmatrix} I_r & 0\ 0 & 0 \end{pmatrix} W^*, ]
and an analogous expression for (VV^*). Hence both associated projections have rank (r), although they act on different ambient spaces when (m\ne n).
An isometry is the special case in which (V^V=I), so its initial space is the entire domain. A coisometry satisfies (VV^=I), making its final space the entire codomain. A unitary operator satisfies both identities and is therefore a partial isometry with zero kernel and surjective range.
Polar decomposition
Every bounded operator (T:H\to K) has a polar decomposition
[ T=V|T|, \qquad |T|=(T^*T)^{1/2}, ]
in which (V) is a partial isometry. Its initial space is
[ \overline{\operatorname{ran}|T|}
(\ker T)^\perp, ]
and its final space is
[ \overline{\operatorname{ran}T}. ]
On the initial space, (V) records the direction of (T), while the positive operator (|T|) records its magnitude. The requirement that (V) vanish on (\ker T) determines it uniquely. Without that requirement, different extensions may act differently on the kernel while producing the same product (V|T|).
The polar decomposition extends the scalar expression (z=u|z|), where (u) has modulus one when (z\ne0) and is set equal to zero when (z=0). In the operator setting, a global unitary factor does not always exist because the closures of the range of (T) and the range of (T^*) need not exhaust their ambient spaces. The partial isometry supplies exactly the isometric component supported by those ranges.
John von Neumann used this operator form in the structural analysis of closed and unbounded operators. Subsequent formulations incorporated the same partial-isometric factor into the theory of von Neumann algebras, where its initial and final projections became invariants of operator-algebraic equivalence.
Defect spaces and index data
The orthogonal complements of the initial and final spaces are called the defect spaces:
[ \mathcal D_{\mathrm{in}}=\ker V, \qquad \mathcal D_{\mathrm{out}}=\ker V^*. ]
Their dimensions measure the failure of (V) to be an isometry on the whole domain and the failure of its range to fill the codomain. When these dimensions are finite, their difference is related to the Fredholm index:
[ \operatorname{ind}V
\dim\ker V-\dim\ker V^*. ]
You Watanabe’s 1956 analysis of partial isometries organized these two defect dimensions as an ordered pair and established their behavior under orthogonal direct sums and compatible compositions. Her formulation distinguished the loss of initial directions from the absence of final directions, avoiding their collapse into the single integer supplied by the Fredholm index. This distinction is relevant in infinite-dimensional settings, where both defect spaces may be nontrivial even when their dimension difference vanishes.
For a finite-dimensional square matrix, the two defect dimensions coincide because the initial and final spaces have the same rank. In infinite-dimensional Hilbert spaces they need not coincide. The unilateral shift on (\ell^2(\mathbb N)), defined by
[ S(x_0,x_1,x_2,\ldots)=(0,x_0,x_1,\ldots), ]
is an isometry satisfying (S^S=I), but (SS^) is the projection onto the subspace of sequences whose first coordinate is zero. Its initial defect space is trivial, whereas its final defect space is one-dimensional.
Partial isometries in operator algebras
Within a C*-algebra (A), an element (v) is a partial isometry when (v^v) is a projection. The element (vv^) is then also a projection, and the relations
[ v^v=p, \qquad vv^=q ]
express that (v) implements an equivalence between (p) and (q). This relation is known as Murray–von Neumann equivalence, following its use by Francis Murray and John von Neumann in the classification of projections in operator algebras.
The equivalence compares projections through the existence of a partial isometry rather than through equality inside the algebra. In the algebra (B(H)) of all bounded operators on a Hilbert space, two projections are Murray–von Neumann equivalent exactly when their ranges have the same Hilbert-space dimension. In a general C*-algebra, the relation contains structural information that is not reducible to ordinary matrix rank and contributes to the construction of K-theory.
Partial isometries also describe the normalizers of certain subalgebras. If (D) is a commutative C*-subalgebra, a partial isometry (v) satisfying appropriate inclusion relations between (vDv^*) and (D) induces a partially defined symmetry of the spectrum of (D). The partial character of the symmetry corresponds to the fact that the initial and final projections need not be the identity.
Composition
The product of two partial isometries is not necessarily a partial isometry. Let (U) and (V) be composable partial isometries. Their product (UV) is a partial isometry precisely when the initial projection (U^U) commutes with the final projection (VV^):
[ (U^U)(VV^)=(VV^*)(U^*U). ]
This condition controls the interaction between the subspace reached by (V) and the subspace on which (U) acts isometrically. When the projections commute, their product is again a projection, which yields the projection identity required for (UV). Without commutation, the intermediate geometry may distort norms on the proposed initial space.
The direct sum of partial isometries is always a partial isometry. For a family ((V_i)) acting between mutually orthogonal Hilbert-space summands,
[ \left(\bigoplus_i V_i\right)^* \left(\bigoplus_i V_i\right)
\bigoplus_i V_i^*V_i, ]
and the right-hand side is an orthogonal projection. This construction preserves the initial spaces, final spaces, and defect spaces as corresponding orthogonal direct sums.