Partial isometry

The concept of partial isometry can be defined in other equivalent ways. If U is an isometric map defined on a closed subset H1 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 H1. 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. == Characterization in finite dimensions ==

In finite-dimensional vector spaces, a matrix A is a partial isometry if and only if A^* A is the projection onto its support. Contrast this with the more demanding definition of isometry: a matrix V is an isometry if and only if V^* V=I. In other words, an isometry is an injective partial isometry. Any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form A=\begin{pmatrix}V & 0\end{pmatrix}, that is, as a matrix whose first \operatorname{rank}(A) columns form an isometry, while all the other columns are identically 0. Note that for any isometry V, the Hermitian conjugate V^* is a partial isometry, although not every partial isometry has this form, as shown explicitly in the given examples. == Operator Algebras ==

For operator algebras, one introduces the initial and final subspaces: :\mathcal{I}W:=\mathcal{R}W^*W,\,\mathcal{F}W:=\mathcal{R}WW^* == C*-Algebras ==

For C*-algebras, one has the chain of equivalences due to the C*-property: :(W^*W)^2 = W^*W \iff WW^*W = W \iff W^*WW^* = W^* \iff (WW^*)^2 = WW^* 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: :P = W^*W,\,Q = WW^* 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: :P:\mathcal{H}\rightarrow\mathcal{H}:\quad\mathcal{I}P=\mathcal{F}P Embeddings Any isometric embedding is one with full initial subspace: :J:\mathcal{H}\hookrightarrow\mathcal{K}:\quad\mathcal{I}J=\mathcal{H} Unitaries Any unitary operator is one with full initial and final subspace: :U:\mathcal{H}\leftrightarrow\mathcal{K}:\quad\mathcal{I}U=\mathcal{H},\,\mathcal{F}U=\mathcal{K} (Apart from these there are far more partial isometries.) == Examples ==

Nilpotents On the two-dimensional complex Hilbert space the matrix :\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix} is a partial isometry with initial subspace : \{0\} \oplus \mathbb{C} and final subspace : \mathbb{C} \oplus \{0\}. Generic finite-dimensional examples Other possible examples in finite dimensions are A\equiv \begin{pmatrix}1&0&0\\0&\frac1{\sqrt2}&\frac1{\sqrt2}\\0&0&0\end{pmatrix}. This is clearly not an isometry, because the columns are not orthonormal. However, its support is the span of \mathbf e_1\equiv (1,0,0) and \frac{1}{\sqrt2}(\mathbf e_2+\mathbf e_3)\equiv (0,1/\sqrt2,1/\sqrt2), and restricting the action of A on this space, it becomes an isometry (and in particular, a unitary). One can similarly verify that A^* A = \Pi_{\operatorname{supp}(A)}, that is, that A^* A is the projection onto its support. Partial isometries do not necessarily correspond to squared matrices. Consider for example, A\equiv \begin{pmatrix}1&0&0\\0&\frac12&\frac12\\ 0 & 0 & 0 \\ 0& \frac12 & \frac12\end{pmatrix}.This matrix has support the span of \mathbf e_1\equiv (1,0,0) and \mathbf e_2+\mathbf e_3\equiv (0,1,1), and acts as an isometry (and in particular, as the identity) on this space. Yet another example, in which this time A acts like a non-trivial isometry on its support, isA = \begin{pmatrix}0 & \frac1{\sqrt2} & \frac1{\sqrt2} \\ 1&0&0\\0&0&0\end{pmatrix}.One can readily verify that A\mathbf e_1=\mathbf e_2, and A \left(\frac{\mathbf e_2 + \mathbf e_3}{\sqrt2}\right) = \mathbf e_1, showing the isometric behavior of A between its support \operatorname{span}(\{\mathbf e_1, \mathbf e_2+\mathbf e_3\}) and its range \operatorname{span}(\{\mathbf e_1,\mathbf e_2\}). Leftshift and Rightshift On the square summable sequences, the operators :R: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (0,x_1,x_2,\ldots) :L: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (x_2,x_3,\ldots) which are related by :R^* = L are partial isometries with initial subspace :LR(x_1,x_2,\ldots)=(x_1,x_2,\ldots) and final subspace: :RL(x_1,x_2,\ldots)=(0,x_2,\ldots). == References ==