mathematical functional analysis, a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its...

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed...

an isometry when its action is restricted onto the support of A {\displaystyle A} , that is, it means that U {\displaystyle U} is a partial isometry. As...

complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator. The polar decomposition for matrices generalizes...

{T}}} is the partial isometry that vanishes on the orthogonal complement of U {\displaystyle U} , and A {\displaystyle A} is the isometry that embeds U...

Moore–Penrose pseudoinverse B+ can be. In that case, the operator B+A is a partial isometry, that is, a unitary operator from the range of T to itself. This can...

decomposition A = V | A | , {\displaystyle A=V|A|,\,} it says that the partial isometry V should lie in M and that the positive self-adjoint operator |A| should...

bounded operator ⁠ M , {\displaystyle \mathbf {M} ,} ⁠ there exist a partial isometry ⁠ U , {\displaystyle \mathbf {U} ,} ⁠ a unitary ⁠ V , {\displaystyle...

belonging to M are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element...

meaning that ee = e and e* = e. Every projection is a partial isometry, and for every partial isometry s, s*s and ss* are projections. If e and f are projections...

{\displaystyle U} such that U † U = I {\displaystyle U^{\dagger }U=I} (a partial isometry), the ensemble { q i , | φ i ⟩ } {\displaystyle \{q_{i},|\varphi _{i}\rangle...

{\displaystyle \{x'_{k}:k<n\}} ). The union of these maps defines a partial isometry ϕ : X → X ′ {\displaystyle \phi :X\to X'} whose domain resp. range...

kernel of P, clearly UP h = 0. But PU h = 0 as well. because U is a partial isometry whose initial space is closure of range P. Finally, the self-adjointness...

and 1 − p are Murray–von Neumann equivalent, i.e., there exists a partial isometry u such that p = uu* and 1 − p = u*u. One can easily generalize this...

In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors...

graphical representations of specific states, unitary operators, linear isometries, and projections in the computational basis | 0 ⟩ , | 1 ⟩ {\displaystyle...

at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and...

{\displaystyle T(x_{1},x_{2},x_{3},\dots )=(x_{2},x_{3},x_{4},\dots ).} T is a partial isometry with operator norm 1. So σ(T) lies in the closed unit disk of the complex...

the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. This means that...

Hines, Peter; Braunstein, Samuel L. (2010). "The Structure of Partial Isometries". In Gay and, Simon; Mackie, Ian (eds.). Semantic Techniques in Quantum...

surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic...

operators is equivalent to finding unitary extensions of suitable partial isometries. Let H {\displaystyle H} be a Hilbert space. A linear operator A {\displaystyle...

V2, K2) be two Stinespring representations of a given Φ. Define a partial isometry W : K1 → K2 by W π 1 ( a ) V 1 h = π 2 ( a ) V 2 h . {\displaystyle...

Neumann equivalent, denoted by p ~ q, if p = vv* and q = v*v for some partial isometry v in M∞(A). It is clear that ~ is an equivalence relation. Define a...

C*-algebras and k-graph C*-algebras are universal C*-algebras generated by partial isometries. The universal C*-algebra generated by a unitary element u has presentation...

group of isometries of X {\displaystyle X} acts by homeomorphisms on ∂ X {\displaystyle \partial X} . This action can be used to classify isometries according...

spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed...

bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the...

variety G/P is a compact homogeneous Riemannian manifold K/(K∩P) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous...

follows: The Gaussian curvature of a surface is invariant under local isometry. A sphere of radius R has constant Gaussian curvature which is equal to...