Partial Isometries Research Articles

Advancements in the generalization of m-isometries: introducing the q-partial-(m, A)-isometry class

In this paper, we extend the concepts of m-partial isometry of order q and ( m , A ) -isometry, building on the contributions of Aouichaoui MA. [A note on partial-A-isometries and some applications. Quaest Math. 2024; 47(3):515–535], Bermúdez et al. [ ( m , A ) -isometries on Hilbert spaces. Linear Algebra Appl. 2018;540:95–111], Sadi A, Mahmoudi F. [ ( m , A ) -partial isometries in semi-Hilbertian spaces. Linear Multilinear Algebra. doi: 10.1080/03081087.2022.2068493], Sid Ahmed OAM, Saddi A. [A-m-Isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436(10):3930–3942], Sid Ahmed OAM. [Generalization of m-partial isometries on a Hilbert space. Int J Pure Appl Math. 2015;104(4):599–619]. Specifically, we introduce and define the class of q-partial- ( m , A ) -isometries, and provide a detailed analysis of their matrix representations. Additionally, we explore their spectral properties.

The Hrushovski property for compact special cube complexes

AbstractWe show that any compact nonpositively curved cube complex embeds in a compact nonpositively curved cube complex where each combinatorial injective partial local isometry of extends to an automorphism of . When is special and the collection of injective partial local isometries satisfies certain conditions, we show that can be chosen to be special and the embedding can be chosen to be a local isometry.

Algebraic Properties of the Semigroup of Partial Isometries of a Finite Chain

This work analyses the algebraic properties of the sub-semi group of partial Isometries (〖DP〗_n) and of order preserving partial isometries (〖ODP〗_n) of a finite chain X_n={1,2,…n}, with a symmetric inverse semigroup I_n defined on it. It also investigates the subsemigroups 〖DP〗_n and 〖ODP〗_n for cycle structure and shows that 〖ODP〗_n is a 0 -E- unitary inverse semigroup.

The equation characterizations of generalized inverses in a ring with involution

In this paper, we study the relation between the generalized inverse properties of an element in a ring with involution and related equations. Mainly, by exploring the existence of the solution in a given set and the expressions of the general solution to these constructed related equation, we obtain a lot of new characterizations of EP elements, partial isometries, SEP elements, Hermitian elements and normal elements.

Double-variable trace maximization for extreme generalized singular quartets of a matrix pair: A geometric method

In this paper, we consider the problem of computing an arbitrary generalized singular value of a Grassman or real matrix pair and a triplet of associated generalized singular vectors. Based on the QR factorization, the problem is reformulated as two novel trace maximization problems, each of which has double variables with unitary constraints or orthogonal constraints. Theoretically, we show that the arbitrarily prescribed extreme generalized singular values and associated triplets of generalized singular vectors can be determined by the global solutions of the constrained trace optimization problems. Then we propose a geometric inexact Newton–conjugate gradient (Newton-CG) method for solving their equivalent trace minimization problems over the Riemannian manifold of all fixed-rank partial isometries. The proposed method can extract not only the prescribed extreme generalized singular values but also associated triplets of generalized singular vectors. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Finally, numerical experiments on both synthetic and real data sets show the effectiveness and high accuracy of our method.

Adversarial Robustness with Partial Isometry.

Despite their remarkable performance, deep learning models still lack robustness guarantees, particularly in the presence of adversarial examples. This significant vulnerability raises concerns about their trustworthiness and hinders their deployment in critical domains that require certified levels of robustness. In this paper, we introduce an information geometric framework to establish precise robustness criteria for l2 white-box attacks in a multi-class classification setting. We endow the output space with the Fisher information metric and derive criteria on the input-output Jacobian to ensure robustness. We show that model robustness can be achieved by constraining the model to be partially isometric around the training points. We evaluate our approach using MNIST and CIFAR-10 datasets against adversarial attacks, revealing its substantial improvements over defensive distillation and Jacobian regularization for medium-sized perturbations and its superior robustness performance to adversarial training for large perturbations, all while maintaining the desired accuracy.

H-Toeplitz operators on the Dirichlet type space

<abstract><p>In this paper, we conducted a study of H-Toeplitz operators on the Dirichlet type space $ \mathfrak{D}_{t} $, which included several aspects. To begin, we established the matrix representation of the H-Toeplitz operator $ S_{\varphi} $ with respect to the orthonormal basis of $ \mathfrak{D}_t $. Subsequently, we characterized the compactness of $ S_{\varphi} $ in terms of the symbol $ \varphi $. Furthermore, we developed a new method to investigate the algebraic properties of H-Toeplitz operators, including self-adjointness, diagonality, co-isometry, partial isometry as well as commutativity.</p></abstract>

Characterization of Directed Graphs Representing C*-Algebra of Complex Matrices

Quantum mechanics is a study that plays a major role in determining the biological intelligence of Artificial Intelligence (AI). Point particle systems in quantum mechanics can be explained using C*-Algebra which is called CAR-algebra. There is a special case in the CAR-algebra which is isomorphic to the C*-algebra of complex matrices. On the other hand, C*-algebras of direct sum of complex matrix spaces is isomorphic to C*-algebra constructed by orthogonal projection and partial isometries operators via the Cuntz-Krieger relations of a directed graph. This article will provide a basis for the relationship between quantum mechanics and graphs through a discussion of the characterization of graphs that can represent C*-algebra of complex matrices. It is found that C*-algebra complex matrices n × n is a directed graph without cycles with n – 1 arrows, a single source, and has n path from the source.

On the monoid of partial isometries of a wheel graph

In this paper, we consider the monoid [Formula: see text] of all partial isometries of a wheel graph [Formula: see text] with [Formula: see text] vertices. Our main objective is to determine the rank of [Formula: see text]. In the process, we also compute the ranks of three notable subsemigroups of [Formula: see text]. We also describe Green’s relations of [Formula: see text] and of its three considered subsemigroups.

Characterization of pre-idempotent Copulas

Abstract Copulas C C for which ( C t C ) 2 = C t C {({C}^{t}C)}^{2}={C}^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ {\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ {\mathcal{ {\mathcal R} }} . Identifying copulas with Markov operators on L 2 {L}^{2} , the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ \sigma -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

Endomorphisms of the Toeplitz algebra

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.

Finite Dimensional Labeled Graph Algebras

Given a directed graph E and a labeling L, one forms the labeled graph algebra by taking a weakly left-resolving labeled space (E, L, B) and considering a generating family of partial isometries and projections. In this paper, we discuss details in the formulation of the algebras, provide examples, and formulate a process that describes the algebra given the graph and a labelling.

On the monoid of partial isometries of a cycle graph

On the monoid of partial isometries of a cycle graph

On normal partial isometries

Abstract Our aim in this paper is to determine when a partially isometric matrix is normal. However, we do not restrict ourselves to the finite-dimensional case and we describe when a partial isometry in ℬ ⁢ ( ℋ ) {\mathcal{B}(\mathcal{H})} satisfies several strong and weak normal properties. In particular, we give elegant characterisations of normal partial isometries on infinite-dimensional Hilbert Spaces in terms of generalized inverses; we collect some spectral properties and explore when an operator in ℬ ⁢ ( ℋ ) {\mathcal{B}(\mathcal{H})} is similar to a normal partial isometry. We close the paper by treating when a partial isometry is hyponormal.

Automatic selfadjoint-ideal semigroups for finite matrices

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B(H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B(H) and the finite rank operators F(H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A⁎=XAY which we believe is a new connection relating the semigroup theory with the theory of operator equations.We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators.In this paper, we focus on a separate Mn(C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A⁎=XAY in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries.

A note on partial-A-isometries and some applications

In this paper, the concepts of partial isometry and A-isometry are generalized. The inspiration for our investigation comes from [4, 10, 20, 27, 28, 29].

On EP matrices, partial isometry matrices and their numerical range

In this paper, some properties of the numerical range of some well-known generalized inverses of matrices, such as Moore-Penrose inverse, Drazin inverse, DMP inverse and CMP inverse, are investigated. In particular, an identification of EP matrices is given via the numerical range. Also, some relations between certain orders and numerical ranges of matrices that are ordered, are presented. Moreover, some results related to the sharp points of the numerical range of a partial isometry matrix are obtained. To illustrate the main results, some numerical examples are also given.

On the circular numerical range of 5-by-5 partial isometries

We prove, in some cases in terms of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center must be the origin. This gives a partial affirmative answer to Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14–35.], for the five-dimensional case.

On the range of TRO-conditional expectations

The purpose of this paper is to discuss the relationship between a TRO T and a sub-TRO S that is the range of a TRO-conditional expectation on T (a ternary corner) by investigating a class D of bounded linear maps on T. We pay particular attention to the case when the TROs contain partial isometries.

On low-dimensional partial isometries

Two statements concerning $n$-by-$n$ partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for $n\leq 4$ but fail starting with $n=5$.