Editing Partial isometry (section)

== Examples ==

=== Nilpotents ===

On the two-dimensional [[complex number|complex]] Hilbert space the matrix

:<math>\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}</math>

is a partial isometry with initial subspace

: <math>\{0\} \oplus \mathbb{C}</math>

and final subspace

: <math>\mathbb{C} \oplus \{0\}.</math>

=== Generic finite-dimensional examples ===
Other possible examples in finite dimensions are
<math display="block">A\equiv \begin{pmatrix}1&0&0\\0&\frac1{\sqrt2}&\frac1{\sqrt2}\\0&0&0\end{pmatrix}.</math>
This is clearly not an isometry, because the columns are not [[orthonormal]]. However, its support is the span of <math>\mathbf e_1\equiv (1,0,0)</math> and <math>\frac{1}{\sqrt2}(\mathbf e_2+\mathbf e_3)\equiv (0,1/\sqrt2,1/\sqrt2)</math>, and restricting the action of <math>A</math> on this space, it becomes an isometry (and in particular, a unitary). One can similarly verify that <math>A^* A = \Pi_{\operatorname{supp}(A)}</math>, that is, that <math>A^* A</math> is the projection onto its support.

Partial isometries do not necessarily correspond to squared matrices. Consider for example,
<math display="block">A\equiv \begin{pmatrix}1&0&0\\0&\frac12&\frac12\\ 0 & 0 & 0 \\ 0& \frac12 & \frac12\end{pmatrix}.</math>This matrix has support the span of <math>\mathbf e_1\equiv (1,0,0)</math> and <math>\mathbf e_2+\mathbf e_3\equiv (0,1,1)</math>, and acts as an isometry (and in particular, as the identity) on this space.

Yet another example, in which this time <math>A</math> acts like a non-trivial isometry on its support, is<math display="block">A = \begin{pmatrix}0 & \frac1{\sqrt2} & \frac1{\sqrt2} \\ 1&0&0\\0&0&0\end{pmatrix}.</math>One can readily verify that <math>A\mathbf e_1=\mathbf e_2</math>, and <math>A \left(\frac{\mathbf e_2 + \mathbf e_3}{\sqrt2}\right) = \mathbf e_1</math>, showing the isometric behavior of <math>A</math> between its support <math>\operatorname{span}(\{\mathbf e_1, \mathbf e_2+\mathbf e_3\})</math> and its range <math>\operatorname{span}(\{\mathbf e_1,\mathbf e_2\})</math>.

=== Leftshift and Rightshift ===

On the square summable sequences, the operators

:<math>R: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (0,x_1,x_2,\ldots)</math>

:<math>L: \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}): (x_1,x_2,\ldots) \mapsto (x_2,x_3,\ldots)</math>

which are related by

:<math>R^* = L</math>

are partial isometries with initial subspace

:<math>LR(x_1,x_2,\ldots)=(x_1,x_2,\ldots)</math>

and final subspace:

:<math>RL(x_1,x_2,\ldots)=(0,x_2,\ldots)</math>.