Orthogonality Preserving Research Articles

Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

On local preservation of orthogonality and its application to isometries

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including ℓ∞n,ℓ1n.

Preservation of Order and Orthogonality on Preduals of Jordan Algebras

Preservation of Order and Orthogonality on Preduals of Jordan Algebras

An unconditionally energy-stable and orthonormality-preserving iterative scheme for the Kohn-Sham gradient flow based model

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n, the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at tn+1/2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.

Di-orthographs of Banach spaces that are not invariant of nonlinear Birkhoff-James orthogonality preservers

In this paper, we study the relationship between two known nonlinear equivalences of normed spaces induced by Birkhoff-James orthogonality. To be precise, we consider the nonlinear equivalences of normed spaces with respect to the structure of Birkhoff-James orthogonality and di-orthographs of them. It is shown that a graph isomorphism between di-orthographs of normed spaces gives rise to a Birkhoff-James orthogonality preserver, but the converse is false in general. A counterexample is constructed in the real two-dimensional setting. Radon planes and a reduced version of di-orthographs play important roles in this direction.

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

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.

Linear orthogonality preservers between function spaces associated with commutative JB⋆-triples

It is known, by Gelfand theory, that every commutative JB∗-triple admits a representation as a space of continuous functions of the formC0T(L)={a∈C0(L):a(λt)=λa(t),∀λ∈T,t∈L},where L is a principal T-bundle and T denotes the unit circle in C. We provide a full technical description of all orthogonality preserving (non-necessarily continuous nor bijective) linear maps between commutative JB∗-triples. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB∗-triples is automatically continuous and bi-orthogonality preserving.

On Birkhoff–James orthogonality preservers between real non-isometric Banach spaces

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results of Koldobsky and of Wójcik. Moreover, in an arbitrary dimension, we construct examples of non-isometric pairs of non-smooth real Banach spaces that admit norm preserving homogeneous bicontinuous Birkhoff–James orthogonality preservers among them.

Condition Monitoring of Mechanical Components Based on MEMED-NLOPE under Multiscale Features

An increasing popularity of researches focuses on the vibration signal with the characteristics of nonstationary, nonlinear, and strong noise interference. A nonlinear dimension and feature reduction method called multiple empirical mode entropy decomposition-nonlocal orthogonal preserving embedding (MEMED-NLOPE) is proposed to implement condition monitoring in this paper. Different from multiple empirical mode decomposition (MEMD), MEMED adopts maximum entropy method, which can directly output the subsignal with the maximum correlation and realize nonlinear dimensionality reduction. Besides, multiscale feature extraction method is used during preprocessing nonlinear data process, which realizes feature reduction. Finally, nonlocal orthogonal preserving embedding algorithm-exponentially weighted moving average (NLOPE-EWMA) realizes the automatic detection of the fault. Taking the laboratory rolling bearing test and naval gun pendulum mechanism test as cases, the effectiveness of MEMED-NLOPE is verified.

Nonlinear Birkhoff–James orthogonality preservers in smooth normed spaces

Let X and Y be smooth normed spaces which are either real and dim⁡X≥3, or infinite dimensional complex, and one of them is reflexive. Then a surjective mapping from X to Y preserves Birkhoff–James orthogonality in both directions if and only if it has the form x↦τ(x)Ux for some surjective linear or conjugate linear isometry U:X→Y and some scalar–valued mapping τ on X. In particular, there exists a surjective mapping from X to Y preserving Birkhoff–James orthogonality in both directions if and only if X and Y are isometrically isomorphic or conjugate isometrically isomorphic. Several illustrative examples and relations with Wigner's theorem are also given.

Nonlinear equivalence of Banach spaces based on Birkhoff-James orthogonality

A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by X∼BJY, if there exists a (possibly nonlinear) bijection between X and Y that preserves Birkhoff-James orthogonality in both directions. It is shown that X≅Y if either X or Y is finite dimensional and X∼BJY, and that ℓp≁BJℓq if 1<p<q<∞. Moreover, if H is a Hilbert space with dim⁡H≥3 and H∼BJX, then H=X. In the two-dimensional case, it turns out that ℓp,q2∼BJℓ22, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.

A geometric approach to Wigner‐type theorems

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

One-parameter semigroups of orthogonality preservers of JB*-algebras

One-parameter semigroups of orthogonality preservers of JB*-algebras

One-parameter groups of orthogonality preservers on C$$^*$$-algebras

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C\(^*\)-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C\(^*\)-algebra. We prove that given a family \(\{T_t: t\in {\mathbb {R}}_0^{+}\}\) of orthogonality preserving bounded linear bijections on a general C\(^*\)-algebra A with \(T_0=Id\), if for each \(t\ge 0,\) we set \(h_t = T_t^{**} (1)\) and we write \(r_t\) for the range partial isometry of \(h_t\) in \(A^{**},\) and \(S_t\) stands for the triple isomorphism on A associated with \(T_t\) satisfying \(h_t^* S_t(x)\) \(= S_t(x^*)^* h_t\), \(h_t S_t(x^*)^* =\) \( S_t(x) h_t^*\), \(h_t r_t^* S_t(x) =\) \(S_t(x) r_t^* h_t\), and \(T_t(x) = h_t r_t^* S_t(x) = S_t(x) r_t^* h_t, \hbox { for all } x\in A,\) the following statements are equivalent: (a) \(\{T_t: t\in {\mathbb {R}}_0^{+}\}\) is a uniformly continuous one-parameter semigroup of orthogonality preserving operators on A; (b) \(\{S_t: t\in {\mathbb {R}}_0^{+}\}\) is a uniformly continuous one-parameter semigroup of surjective linear isometries (i.e., triple isomorphisms) on A (and hence there exists a triple derivation \(\delta \) on A such that \(S_t = e^{t \delta }\) for all \(t\in {\mathbb {R}}\)), the mapping \(t\mapsto h_t \) is continuous at zero, and the identity \( h_{t+s} = h_t r_t^* S_t^{**} (h_s),\) holds for all \(s,t\in {\mathbb {R}}.\)

Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.

Gradient Flow Based Kohn--Sham Density Functional Theory Model

In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gradient flow based model. We prove that our gradient flow based model is orthogonality preserving, the extended gradient has an exponential decay over time $t$, and the equilibrium point is a local minimizer of the Kohn--Sham energy functional. Then we propose a midpoint scheme to carry out the temporal discretization, which is proven to be orthogonality preserving, too. Based on the midpoint scheme, we design a practical orthogonality preserving iteration scheme which can deal with the propagation of the gradient flow based model and prove that the scheme produces approximations that converge to a local minimizer with some convergence rate under some reasonable assumptions. Finally, we report a number of numerical experiments that validate our theoretical results.

Complete orthogonality preservers between C⁎-algebras

We introduce n-orthogonality (and completely orthogonality) preserving operators between C⁎-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C⁎-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.

On symmetry of Birkhoff orthogonality in the positive cones of C⁎-algebras with applications

In this paper, local symmetry of Birkhoff orthogonality is considered in the positive cones of C ⁎ -algebras. For the positive cone A + of a C ⁎ -algebra A , the notion of A + -local left (or right) symmetric points for Birkhoff orthogonality is introduced. Some simple characterizations of them are given. As an application, we determine the form of Birkhoff orthogonality preservers between the positive cones of C ⁎ -algebras by using Jordan ⁎-isomorphisms.