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

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...

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...

{\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...

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...

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

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...

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

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...

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...

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

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

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

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...

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...

{\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...

structure of T {\displaystyle T} means that a "truncated" shift is a partial isometry on H {\displaystyle {\mathcal {H}}} . More specifically, let { e 0...

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...

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

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

introduced the notion of a "restricted linear isometry," which is a matrix that is quantitatively close to an isometry when restricted to certain subspaces.[CT05]...

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...

Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The...

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...

ETF ≠ 0 for some T in M. ⇒ ETF has polar decomposition UH for some partial isometry U and positive operator H in M. ⇒ Ran(U) = Ran(ETF) ⊂ Ran(E). Also...

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...