Partial isometry

A partial isometric view is a special type of the mathematical branch of functional analysis examined operators . These are operators that behave like an isometry on a sub-vector space and are otherwise 0, which explains their name. Using partial isometrics, equivalences of projections are defined.

definition

Let be a Hilbert space and a continuous linear operator. is called a partial isometry if the restriction of to the orthogonal complement of is an isometry, i.e. H. . ${\displaystyle H}$ ${\displaystyle U:H\rightarrow H}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle \ker(U)}$ ${\displaystyle \forall x\in \ker(U)^{\perp }:\,\,\|Ux\|=\|x\|}$

The orthogonal complement of the kernel of a partial isometry is called its initial space, the image of a partial isometry is called its target space (final space). Accordingly, a partial isometry is an isometry between its starting space and its target space.

Examples

• Isometrics (especially also unitary operators ) are partial isometries with the special feature that . ${\displaystyle \ker(U)=\{0\}}$
• Orthogonal projections are partial isometries with the special feature that the isometric part, i.e. H. the restriction of orthogonal projection to the orthogonal complement of its core, which is identity.
• ${\displaystyle U={\begin{pmatrix}0&0&0&0\\1&0&0&0\\0&1&0&0\\0&0&0&0\end{pmatrix}}:{\mathbb {C} }^{4}\rightarrow {\mathbb {C} }^{4}}$is a partial isometry with a starting space and a target space . In this example the target space is inclined to the decomposition core + initial space . ${\displaystyle \{(z_{1},z_{2},0,0);z_{i}\in {\mathbb {C} }\}}$ ${\displaystyle \{(0,z_{1},z_{2},0);z_{i}\in {\mathbb {C} }\}}$

properties

If there is a partial isometry, then is the starting space , is the target space. ${\displaystyle U}$ ${\displaystyle \mathrm {im} (U^{*}U)}$ ${\displaystyle \mathrm {im} (UU^{*})}$

For a continuous, linear operator on a Hilbert space, the following statements are equivalent: ${\displaystyle U}$

• ${\displaystyle U}$ is a partial isometry.
• ${\displaystyle U^{*}U}$ is a projection.
• ${\displaystyle U=UU^{*}U}$

With is a partial isometric view, where the start and finish area are exchanged. ${\displaystyle U}$ ${\displaystyle U^{*}}$

Equivalence of projections

Let it be a Von Neumann algebra , i. H. there is a Hilbert space such that there is a C * algebra that matches its bicommutant (see bicommutant theorem ). Two orthogonal projections and from are called equivalent (with respect to ) and one writes if there is a partial isometry with a starting space and a target space , that is, in formulas and . One writes further , if is equivalent to a subprojection of Q, that is, if there is a projection with and . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {A}}\subset L(H)}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle P\sim Q}$ ${\displaystyle U\in {\mathcal {A}}}$ ${\displaystyle \mathrm {im} (P)}$ ${\displaystyle \mathrm {im} (Q)}$ ${\displaystyle P=U^{*}U}$ ${\displaystyle Q=UU^{*}}$ ${\displaystyle P\precsim Q}$ ${\displaystyle P}$ ${\displaystyle P'}$ ${\displaystyle P\sim P'}$ ${\displaystyle P'=P'Q=QP'}$

One can show that there is an equivalence relation on the set of all projections of , and that defines a partial order on the set of equivalence classes. Furthermore, is equivalent to and . This order relation plays an important role in the type classification of Von Neumann algebras. ${\displaystyle \sim }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \precsim }$ ${\displaystyle P\,\sim Q}$ ${\displaystyle P\precsim Q}$ ${\displaystyle Q\precsim P}$

See also

Partial isometrics play an important role in the polar decomposition of operators.

swell

• Paul Halmos : A Hilbert Space Problem Book , Springer-Verlag, ISBN 0387906851
• VS Sunder: An Invitation to Von Neumann Algebras (1987), ISBN 0387963561