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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
Partial_isometry
Distance-preserving mathematical transformation
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed
Isometry
Mathematical study of linear operators
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
Operator_theory
Type of matrix representation
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
Polar_decomposition
Graphical language for quantum processes
graphical representations of specific states, unitary operators, linear isometries, and projections in the computational basis | 0 ⟩ , | 1 ⟩ {\displaystyle
ZX-calculus
Matrix B such that B² equals a given matrix A
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
Square_root_of_a_matrix
Matrix decomposition
bounded operator M {\displaystyle \mathbf {M} } , there exist a partial isometry U {\displaystyle \mathbf {U} } , a unitary V {\displaystyle \mathbf
Singular_value_decomposition
Semigroup in abstract algebra
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
Semigroup_with_involution
*-algebra of bounded operators on a Hilbert space
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
Von_Neumann_algebra
{\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
Urysohn_universal_space
Idempotent linear transformation from a vector space to itself
{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
Projection_(linear_algebra)
structure of T {\displaystyle T} means that a "truncated" shift is a partial isometry on H {\displaystyle {\mathcal {H}}} . More specifically, let { e 0
Trigonometric_moment_problem
Mathematical tool in quantum physics
{\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
Density_matrix
Operation on self-adjoint operators
operators is equivalent to finding unitary extensions of suitable partial isometries. Let H {\displaystyle H} be a Hilbert space. A linear operator A {\displaystyle
Extensions of symmetric operators
Extensions_of_symmetric_operators
Theorem
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
Stinespring_dilation_theorem
Smooth manifold with an inner product on each tangent space
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
Riemannian_manifold
Mathematical ring whose elements are matrices
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
Matrix_ring
Theorem in manifold theory
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
Gauss's lemma (Riemannian geometry)
Gauss's_lemma_(Riemannian_geometry)
Matrix property in linear algebra
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors
Restricted_isometry_property
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
conclude that ∂ x {\displaystyle \partial _{x}} is a Killing field using one of the results below in this article. The isometry group of the upper half-plane
Killing_vector_field
quasinormal if and only if in its polar decomposition A = UP, the partial isometry U and positive operator P commute. Given a quasinormal A, the idea
Subnormal_operator
One-dimensional complex manifold
The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry: genus 0 – the isometry group
Riemann_surface
Concept in mathematics
group of isometries of X {\displaystyle X} acts by homeomorphisms on ∂ X {\displaystyle \partial X} . This action can be used to classify isometries according
Hyperbolic_metric_space
Differential operator in mathematics
{1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}
Laplace_operator
Australian and American mathematician (born 1975)
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]
Terence_Tao
Mathematics of smooth surfaces
Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional spaces by Riemann
Differential geometry of surfaces
Differential_geometry_of_surfaces
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
Quasinormal_operator
{\displaystyle H} . By the polar decomposition theorem, there exists a unique partial isometry U {\displaystyle U} such that T = U | T | {\displaystyle T=U|T|} and
Aluthge_transform
Construction in functional analysis, useful to solve differential equations
{\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
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
C*-algebra
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
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
Structure in group theory (in mathematics)
Hines, Peter; Braunstein, Samuel L. (2010). "The Structure of Partial Isometries". In Gay and, Simon; Mackie, Ian (eds.). Semantic Techniques in Quantum
Inverse_semigroup
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
Affiliated_operator
Probability problem
this motivates Krein's formula which parametrizes the extensions of partial isometries. The cumulative distribution function and the probability density
Hamburger_moment_problem
Calculus of stochastic differential equations
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
Itô_calculus
Mathematical property
F" means that E and F are the initial and final projections of some partial isometry in the algebra (that is, E = V*V and F = VV* for some V in the algebra)
Schröder–Bernstein_property
Product of the principal curvatures of a surface
surface S in R3. A local isometry is a diffeomorphism f : U → V between open regions of R3 whose restriction to S ∩ U is an isometry onto its image. Theorema
Gaussian_curvature
Result of differential geometry proved by Gauss
V → V ~ {\displaystyle \phi :V\to {\tilde {V}}} is an isometry. If there exists local isometries for each p ∈ S {\displaystyle p\in S} then S {\displaystyle
Theorema_Egregium
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
Central_carrier
Feature of a system that is preserved under some transformation
spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed
Symmetry_(physics)
_{\mathbb {C} }^{n}\cup \partial \mathbb {H} _{\mathbb {C} }^{n}} . By Brouwer's fixed point theorem, any holomorphic isometry of the complex hyperbolic
Complex_hyperbolic_space
No complete regular surface of constant negative gaussian curvature immerses in R3
infinite. Proof's Sketch: The idea of the proof is to create a global isometry between H {\displaystyle H} and S ′ {\displaystyle S'} . Then, since H
Hilbert's theorem (differential geometry)
Hilbert's_theorem_(differential_geometry)
Mathematical operation
}x^{-{\frac {1}{2}}-is}\varphi (s)\,ds.} Furthermore, this operator is an isometry, that is to say ‖ M ~ f ‖ L 2 ( − ∞ , ∞ ) = ‖ f ‖ L 2 ( 0 , ∞ ) {\displaystyle
Mellin_transform
Function, homomorphism, or morphism
also been given specific names. These include homomorphisms in algebra, isometries in geometry, operators in analysis and representations in group theory
Map_(mathematics)
e : e ∈ E 1 } {\displaystyle \left\{s_{e}:e\in E^{1}\right\}} are partial isometries with mutually orthogonal ranges, the elements of { p v : v ∈ E 0 }
Graph_C*-algebra
Type of mathematical space
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
Generalized_flag_variety
Model of n-dimensional hyperbolic geometry
Minkowski bilinear form. In a different language, it is the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid
Hyperboloid_model
{\displaystyle X} to be the origin. A geodesic ray is a path given by an isometry γ : [ 0 , ∞ ) → X {\displaystyle \gamma :[0,\infty )\rightarrow X} such
Gromov_boundary
M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that
Schröder–Bernstein theorems for operator algebras
Schröder–Bernstein_theorems_for_operator_algebras
Bijection of a set using properties of shapes in space
Displacements preserve distances and oriented angles (e.g., translations); Isometries preserve angles and distances (e.g., Euclidean transformations); Similarities
Geometric_transformation
Mathematical space with a notion of distance
bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the
Metric_space
an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from ultragraphs.pp
Ultragraph_C*-algebra
Theory of supergravity in four dimensions
undo the isometry transformation, defined by ξ I m ∂ m K + ξ I n ¯ ∂ n ¯ K = r I ( ϕ ) + r ¯ I ( ϕ ¯ ) . {\displaystyle \xi _{I}^{m}\partial _{m}K+\xi
4D_N_=_1_supergravity
Metric tensor describing constant negative (hyperbolic) curvature
The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the
Poincaré_metric
Lie group of Lorentz transformations
Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave a single point (event)
Lorentz_group
Unit-distance-preserving maps are isometries
homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named after Frank S. Beckman and Donald A
Beckman–Quarles_theorem
Relativistic wave equation in quantum mechanics
Lorentz group. Together, these form the Poincaré group which encodes the isometries of flat spacetime. Scalar fields transform as scalars under Lorentz transformations
Klein–Gordon_equation
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
Universal_C*-algebra
2006). "Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching". Proc. Natl. Acad. Sci. U.S.A. 103 (5): 1168–72
Generalized multidimensional scaling
Generalized_multidimensional_scaling
works only in dimension 2). Almost flat manifold Arc-wise isometry the same as path isometry. Asymptotic cone Autoparallel the same as totally geodesic
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Type of vector space in math
that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier
Hilbert_space
there exists an "almost isometry" between Y ′ {\displaystyle Y'} and Y {\displaystyle Y} with respect to which the (partial) actions of B {\displaystyle
Outer_space_(mathematics)
Mathematical problem
dimensional space) to itself that preserves unit distances must be an isometry, preserving all distances. Finite colorings of these spaces can be used
Hadwiger–Nelson_problem
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Solution of Einstein field equations
{\displaystyle -2\exp(-x)\,\partial _{t}+y\,\partial _{x}+\left(\exp(-2x)-y^{2}/2\right)\,\partial _{y}.} The isometry group acts 'transitively' (since we can
Gödel_metric
Geometrical construct in general relativity
thermal radiation and spacetimes that admit a one-parameter group of isometries possessing a bifurcate Killing horizon, which consists of a pair of intersecting
Killing_horizon
Canadian-American mathematician (1925–2020)
mathematicians of the 20th century. Nearly all of his work was in the field of partial differential equations. Many of his contributions are now regarded as fundamental
Louis_Nirenberg
Supergravity in eleven dimensions
Standard Model gauge group, assuming that this arises as subgroup of the isometry group of the compact manifold. The main area of study was understanding
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
Multi-dimensional generalization of triangle
v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}} by the affine isometry that sends v 0 {\displaystyle \scriptstyle v_{0}} to v 0 {\displaystyle
Simplex
Exterior algebraic map taking tensors from p forms to n-p forms
{\partial C}{\partial y}}-{\frac {\partial B}{\partial z}},\,-{\frac {\partial C}{\partial x}}+{\frac {\partial A}{\partial z}},\,{\frac {\partial B}{\partial
Hodge_star_operator
Property of objects which appear unchanged after a partial rotation
rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational
Rotational_symmetry
in H−k(T2) and Pk annihilates C∞ c(Ωc). Canonical isometries: The operator (I + ∆)k gives an isometry of H 2k 0(Ω) into H0(Ω) and of H k 0(Ω) onto H−k(Ω)
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Polygon with an infinite number of sides
space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping. Generally, the moduli space of a faithful
Apeirogon
Extension of Poincaré spacetime symmetry
conformal transformations. Formally, Poincaré transformations are the isometries of Minkowski spacetime, meaning that they preserve the Minkowski metric
Conformal_symmetry
Geometric surface
the tractrices that generate the pseudosphere. This mapping is a local isometry, and thus exhibits the portion y ≥ 1 of the upper half-plane as the universal
Pseudosphere
Stochastic process modeling random walk with friction
{\displaystyle {\frac {\partial P}{\partial t}}=\theta {\frac {\partial }{\partial x}}((x-\mu )P)+D{\frac {\partial ^{2}P}{\partial x^{2}}}} where D = σ
Ornstein–Uhlenbeck_process
Geometric system used in black hole physics
of a black hole. A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3)
Spherically symmetric spacetime
Spherically_symmetric_spacetime
In mathematics, invertible homomorphism
depending on the type of structure under consideration. For example: An isometry is an isomorphism of metric spaces. A homeomorphism is an isomorphism of
Isomorphism
French mathematical physicist (1923–2025)
containing f1(M) and an open subset U2 of M2 containing f2(M), together with an isometry i : (U1, g1) → (U2, g2) such that i(f1(p)) = f2(p) for all p in M. In a
Yvonne_Choquet-Bruhat
Mathematical function that preserves angles
an isometry, and a special conformal transformation. For linear transformations, a conformal map may only be composed of homothety and isometry, and
Conformal_map
Mathematical manifold theory
that the image of the isometry group of M in the general linear group GL(H∗(M, Z)) is finite (because the group of isometries of a lattice is finite)
Hodge_theory
Tensor field in Riemannian geometry
and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space). Since the Levi-Civita
Riemann_curvature_tensor
Mathematical description of spacetime used in relativity
from the three spatial dimensions. In 3-dimensional Euclidean space, the isometry group (maps preserving the regular Euclidean distance) is the Euclidean
Minkowski_spacetime
symmetries. Namely, our spacetime admits a six-dimensional Lie group of self-isometries. This group is generated by a six-dimensional Lie algebra of Killing vector
Monochromatic electromagnetic plane wave
Monochromatic_electromagnetic_plane_wave
Function's sensitivity to argument change
exactly one (which can only happen if A is a scalar multiple of a linear isometry), then a solution algorithm can find (in principle, meaning if the algorithm
Condition_number
Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes,
Spectral_shape_analysis
Way to divide polygon into smaller parts
f:R^{n}(X)\rightarrow S_{R}} . Subdivision rules can be used to study the quasi-isometry properties of certain spaces. Given a subdivision rule R {\displaystyle
Finite_subdivision_rule
Mathematics award
groundbreaking work on Ricci limit spaces, in particular rectifiability, isometry group and co-dimension 4 conjecture." 2025 Vesselin Dimitrov "for major
Fermat_Prize
Scientific theory
{\partial }{\partial t}}{\frac {\partial }{\partial \tau }}\mathbb {E} (W_{t}W_{\tau })={\frac {\partial }{\partial t}}{\frac {\partial }{\partial \tau
Langevin_dynamics
Matrix of inner products of vectors
\mathbb {R} ^{k}} (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence of vectors results in the same Gram matrix
Gram_matrix
Manifold with inversion symmetry
manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces
Hermitian_symmetric_space
Fractal composed of tangent circles
sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry. The Apollonian gasket is the limit set of a group of Möbius transformations
Apollonian_gasket
Russian-French mathematician
application of Gromov and Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Rigidity theorem in differential geometry
theorem, upon viewing the Gauss–Codazzi equations as a system of first-order partial differential equations for the two coordinate derivatives of the position
Bonnet_theorem
Stochastic process generalizing Brownian motion
Gaussian white noise G {\displaystyle G} (with Lebesgue intensity) as an isometry from the space of square-integrable functions L 2 ( R , B ( R ) ) {\displaystyle
Wiener_process
Type of derivative in differential geometry
\psi } can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression
Lie_derivative
(h)} is a parabolic isometry of H 3 {\displaystyle \mathbb {H} ^{3}} if and only if h {\displaystyle h} is a parabolic isometry of H 2 {\displaystyle
Cannon–Thurston_map
Japanese mathematician (1915–2008)
October 2008. Itô calculus Itô diffusion Itô integral Itô–Nisio theorem Itô isometry Itô's lemma Black–Scholes model O'Connor, John J.; Robertson, Edmund F
Kiyosi_Itô
Field theory of scalar fields
for some function λ(x). The conformal group contains as subgroups the isometries of the metric η μ ν {\displaystyle \eta _{\mu \nu }} (the Poincaré group)
Scalar_field_theory
Maximally symmetric Lorentzian manifold with a positive cosmological constant
Topologically, dSn is R × Sn−1, which is simply connected if n ≥ 3. The isometry group of de Sitter space is the Lorentz group O(1, n). The metric therefore
De_Sitter_space
Invariant measure of fractal dimension
condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a
Hausdorff_dimension
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PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
PARTIAL ISOMETRY
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