Linear Isometry Research Articles

Isometries of Lipschitz‐free Banach spaces

AbstractWe describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3‐connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz‐free rigid space that has only three more points.

Optimal pinwheel partitions for the Yamabe equation

Abstract We establish the existence of an optimal partition for the Yamabe equation in R N made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to − ∞ , the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in R N that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.

Linear isometries of [formula omitted]

A complete characterisation is given of all the linear isometries of the Fréchet space of all holomorphic functions on the unit disc, when it is given one of the two standard metrics: these turn out to be weighted composition operators of a particular form. Operators similar to an isometry are also classified. Further, the larger class of operators isometric when restricted to one of the defining seminorms is identified. Finally, the spectra of such operators are studied.

Group-Invariant Max Filtering

AbstractGiven a real inner product space V and a group G of linear isometries, we construct a family of G-invariant real-valued functions on V that we call max filters. In the case where $$V={\mathbb {R}}^d$$ V = R d and G is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $$V=L^2({\mathbb {R}}^d)$$ V = L 2 ( R d ) and G is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.

Homotopical foundations of parametrized quantum spin systems

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space [Formula: see text] to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space [Formula: see text] at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to [Formula: see text]-spaces for an operad we call the “multiplicative” linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev’s loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected.

On Coarse Isometries and Linear Isometries between Banach Spaces

Let X,Y be two Banach spaces and f:X→Y be a standard coarse isometry. In this paper, we first show a sufficient and necessary condition for the coarse left-inverse operator of general Banach spaces to admit a linearly isometric right inverse. Furthermore, by using the well-known simultaneous extension operator, we obtain an asymptotical stability result when Y is a space of continuous functions. In addition, we also prove that every coarse left-inverse operator does admit a linear isometric right inverse without other assumptions when Y is a Lp(1<p<∞) space, or both X and Y are finite dimensional spaces of the same dimension. Making use of the results mentioned above, we generalize several results of isometric embeddings and give a stability result of coarse isometries between Banach spaces.

2-LOCAL ISOMETRIES OF SOME NEST ALGEBRAS

AbstractLet H be a complex separable Hilbert space with $\dim H \geq 2$ . Let $\mathcal {N}$ be a nest on H such that $E_+ \neq E$ for any $E \neq H, E \in \mathcal {N}$ . We prove that every 2-local isometry of $\operatorname {Alg}\mathcal {N}$ is a surjective linear isometry.

ISOMETRIES AND HERMITIAN OPERATORS ON SPACES OF VECTOR-VALUED LIPSCHITZ MAPS

Abstract We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\operatorname {Lip}(X,E)$ , where E is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $\operatorname {Lip}(X,\mathcal {A})$ , where $\mathcal {A}$ is a unital factor $C^{*}$ -algebra. These results improve previous results stated by the author.

An Algebraic Geometric Foundation for a Classification of Second-Order Superintegrable Systems in Arbitrary Dimension

Second-order (maximally) superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem—based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying those superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions.

MacWilliams' Extension Theorem for rank-metric codes

The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.

Secondary-school teachers’ noticing of aspects of mathematics teaching talk in the context of one-day workshops

In a research project with one-day teacher education workshops for secondary-school mathematics teachers, our study explores the potential of tool-supported discussions in helping them to notice important and critical aspects of mathematics teaching talk. Mathematical practices of naming and explaining in teaching talk, students’ content learning challenges, and noticing processes of identifying, interpreting and deciding are the components of our framework and the tools that guided the design and implementation of three workshops on linear equations, fractions and plane isometries. The data was collected during the discussions with the seven teachers and the teacher educator throughout these workshops. The coding of the discussions allowed us to see discourse moves that reveal the teachers’ noticing of: (i) challenges in the identification of mathematical naming, (ii) mathematical explaining that voices the students’ learning, (iii) classroom practice in relation to mathematical naming and explaining.

Phase-isometries on the unit sphere of CL-spaces

A mapping f:SX→SY between the unit spheres of two real Banach spaces X and Y is said to be a phase-isometry if it satisfies{‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖} for all x,y∈SX. When f is surjective, X is a real CL-space and Y is an arbitrary real Banach space, we establish in this paper that there exists a phase-function ε:SX→{−1,1} such that ε⋅f is an isometry which is the restriction of a linear isometry from X to Y.

MIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES

AbstractSuppose that X and Y are two real normed spaces. A map $f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies $$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$ We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, $f(x)=\varepsilon (x)g(x)$ for $x\in X$ , where $g:X\rightarrow Y$ is a linear isometry and $\varepsilon $ is a map from X to $\{-1,1\}$ .

Surjective isometries on the Banach algebra of continuously differentiable maps with values in Lipschitz algebra

Let $${\text {Lip}}(I)$$ be the Banach algebra of all Lipschitz functions on the closed unit interval I with the norm $$\Vert f\Vert _L=\Vert f\Vert _\infty +L(f)$$ for $$f\in {\text {Lip}}(I)$$ , where L(f) is the Lipschitz constant of f. We denote by $$C^{1}(I, {\text {Lip}}(I))$$ the Banach algebra of all continuously differentiable functions F from I to $${\text {Lip}}(I)$$ equipped with the norm $$\Vert F\Vert _{\Sigma }=\sup _{s\in I}\Vert F(s)\Vert _L+\sup _{t\in I}\Vert D(F)(t)\Vert _L$$ for $$F\in C^{1}(I, {\text {Lip}}(I))$$ . In this paper, we prove that if T is a surjective, not necessarily linear, isometry on $$C^{1}(I, {\text {Lip}}(I))$$ , then $$T-T(0)$$ is a weighted composition operator or its complex conjugation. Among other things, any surjective complex linear isometry on $$C^{1}(I, {\text {Lip}}(I))$$ is of the following form: $$c_{1}F(\tau _1(s),\tau _2(x))$$ , where $$c_{1}$$ is a complex number of modulus 1, and $$\tau _1$$ and $$\tau _2$$ are isometries of I onto itself.

Surjective isometries on a Banach space of analytic functions with bounded derivatives

Let $$H(\mathbb D)$$ be the linear space of all analytic functions on the open unit disc $$\mathbb D$$ and $$H^p(\mathbb D)$$ the Hardy space on $$\mathbb D$$ . The characterization of complex linear isometries on $$\mathcal {S}^p=\{ f\in H(\mathbb D):f'\in H^p(\mathbb D) \}$$ was given for $$1 \le p < \infty $$ by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on $$\mathcal {S}^\infty $$ .

On the Structure of Coisometric Extensions

If T is a bounded linear operator on a Hilbert space H and V is a given linear isometry on a Hilbert space K, we present necessary and sufficient conditions on T in order to ensure the existence of a linear isometry π:H→K such that πT=V*π (i.e., (π,V*) extends T). We parametrize the set of all solutions π of this equation. We show, for example, that for a given unitary operator U on a Hilbert space E and for the multiplication operator by the independent variable Mz on the Hardy space HD2(D), there exists an isometric operator π:H→E⊕HD2(D) such that (π,(U⊕Mz)*) extends T if and only if T is a contraction, the defect index δT≤dimD and, for some Y:AT→E, (Y,U*) extends the isometric operator AT1/2h↦AT1/2Th on the space AT=ATH¯, where AT is the asymptotic limit associated with T. We also prove that if T is isometric and V is unitary, there exists an isometric operator π:H→K such that (π,V) extends T if and only if (a) the spectral measures of the unitary part of T (in its Wold decomposition) and the restriction of V to one of its reducing subspaces K0 possess identical multiplicity functions and (b) dim(kerT*)=dim(K1⊖VK1) for a certain subspace K1 of K that contains K0 and is invariant under V. The precise form of π, in each situation, and characterizations of the minimality conditions are also included. Several examples are given for illustrative purposes.

Preservers of Triple Transition Pseudo-Probabilities in Connection with Orthogonality Preservers and Surjective Isometries

We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW^*-triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projections in B(H). We also present a Tingley type theorem by proving that every surjective isometry between the sets of minimal tripotents in two atomic JBW^*-triples admits an extension to a real linear surjective isometry between these two JBW^*-triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW^*-triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.

Filament plots for data visualization

The efficiency of modern computer graphics allows us to explore collections of space curves simultaneously with “drag-to-rotate” interfaces. This inspires us to replace “scatterplots of points” with “scatterplots of curves” to simultaneously visualize relationships across an entire dataset. Since spaces of curves are infinite dimensional, scatterplots of curves avoid the “lossy” nature of scatterplots of points. In particular, if two points are close in a scatterplot of points derived from high-dimensional data, it does not generally follow that the two associated data points are close in the data space. Andrews plots provide scatterplots of curves that perfectly preserve Euclidean distances, but simultaneous visualization of these graphs over an entire dataset produces “visual clutter” because graphs of functions generally overlap in 2D. We mitigate this “visual clutter” issue by constructing computationally inexpensive 3D extensions of Andrews plots. First, we construct optimally smooth 3D Andrews plots by considering linear isometries from Euclidean data spaces to spaces of planar parametric curves. We rigorously parametrize the linear isometries that produce (on average) optimally smooth curves over a given dataset. This parameterization of optimal isometries reveals many degrees of freedom, and (using recent results on generalized Gauss sums) we identify a particular member of this set which admits an asymptotic “tour” property that avoids certain local degeneracies as well. Finally, we construct unit-length 3D curves (filaments) from Bishop frames induced by 3D Andrews plots. We conclude with examples of filament plots for several standard datasets, 1 illustrating how filament plots avoid “visual clutter”.

Every commutative JB^*-triple satisfies the complex Mazur–Ulam property

We prove that every commutative JB^*-triple, represented as a space of continuous functions C_0^{mathbb {T}}(L), satisfies the complex Mazur–Ulam property, that is, every surjective isometry from the unit sphere of C_0^{mathbb {T}}(L) onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.

Saturated and linear isometric transfer systems for cyclic groups of order pmqn

Transfer systems are combinatorial objects which classify N∞ operads up to homotopy. By results of A. Blumberg and M. Hill [2], every transfer system associated to a linear isometries operad is also saturated (closed under a particular two-out-of-three property). We investigate saturated and linear isometric transfer systems with equivariance group Cpmqn, the cyclic group of order pmqn for p,q distinct primes and m,n≥0. We give a complete enumeration of saturated transfer systems for Cpmqn. We also prove J. Rubin's saturation conjecture for Cpqn; this says that every saturated transfer system is realized by a linear isometries operad for p,q sufficiently large (greater than 3 in this case).