Examples Nilpotents On the two-dimensional complex Hilbert space the matrix is a partial isometry with initial subspace and final subspace Generic finite-dimensional examples Other possible examples in finite dimensions areThis is clearly not an isometry, because the columns are not orthonormal. However, its support is the span of and , and restricting the action of on this space, it becomes an isometry (and in particular a unitary). One can similarly verify that , that is, that is the projection onto its support. Partial isometries need not correspond to squared matrices. Consider for example,This matrix has support the span of and , and acts as an isometry (and in particular, as the identity) on this space. Yet another example, in which this time acts like a non-trivial isometry on its support, isOne can readily verify that , and , showing the isometric behavior of between its support and its range . Leftshift and Rightshift On the square summable sequences the operators which are related by are partial isometries with initial subspace and final subspace: . References John B. Conway (1999). "A course in operator theory", AMS Bookstore, Alan L. T. Paterson (1999). "Groupoids, inverse semigroups, and their operator algebras", Springer, Mark V. Lawson (1998). "Inverse semigroups: the theory of partial symmetries". World Scientific External links Important properties and proofs Alternative proofs Operator theory C*-algebras Semigroup theory