Partial isometry

A partial isometry is a special type studied in the mathematical branch of functional analysis operators. Here are operators on a subspace behave like an isometry and otherwise 0, which explains its name. By means of partial isometries are defined equivalences of projections.

Definition

Be a Hilbert space and a continuous linear operator. is a partial isometric view when the restriction of the orthogonal complement of an isometric view, that is,.

The orthogonal complement of the core of a partial isometry is called its initial space ( engl. initial space), the image of a partial isometry is called its target space (English final space). Thus, a partial isometry is an isometry between its initial space and objective space.

Examples

• Isometries ( specifically including unitary operators ) are partial isometries with the peculiarity that.
• Orthogonal projections are partial isometries with the particularity that the isometric portion, ie the restriction of the orthogonal projection onto the orthogonal complement its core, is the identity.
• Is a partial isometry with initial space and target space. In this example, the target space is oblique to the decomposition core top space.

Properties

Is a partial isometry, so is the initial space, the target space.

For a steady, linear operator on a Hilbert space the following are equivalent:

• Is a partial isometry.
• Is a projection.

With is also a partial isometry, where the start and finish area are replaced.

Equivalence of projections

There is a von Neumann algebra, i.e. there is a Hilbert space, so that a C *-algebra, which coincides with their Bikommutanten (see Bikommutantensatz ). Two orthogonal projections and out are called equivalent (with respect to ) and we write, if there is a partial isometry with initial space and target space, ie in formulas and. Further written as if equivalent to a lower projection of Q, that is, when there is a projection.

One can show that an equivalence relation on the set of all projections of, and that defines a partial order on the set of equivalence classes. Furthermore, equivalent to and. This order relation plays an important role in the type classification of von Neumann algebras.