Search Results (95)

In this paper, we introduce and investigate the concept of K-g-fusion frames in the Cartesian product of two Hilbert $C^{*}$-modules over the same unital $C^{*}$-algebra. Our main result establishes that the Cartesian product of two K-g-fusion frames remains a K-g-fusion frame for the direct-sum module. We give explicit formulae for the associated synthesis, analysis, and frame operators and prove natural relations (direct-sum decomposition of the frame operator). Furthermore, we prove a perturbation theorem showing that small perturbations of the component families, measured in the operator or norm sense, still yield a K-g-fusion frame for the product module, with explicit new frame bounds obtained. Full article

This manuscript introduces and investigates a new class of operators, termedn-quasi-$[m,\mathcal{C}]$-symmetric operators, which generalize and extend the existing notions of $[m,\mathcal{C}]$-symmetric and n-quasi-$[m,\mathcal{C}]$-isometric operators. Specifically, given a conjugation $\mathcal{C}$ on a Hilbert space, an operator $\mathcal{Q}\in \mathcal{B}\left(\mathcal{K}\right)$ is said to be n-quasi-$[m,\mathcal{C}]$-symmetric if it satisfies the relation${\mathcal{Q}}^{\ast n}\left({\displaystyle \sum _{0\le j\le m}}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{m}{j}}\right)\mathcal{C}{\mathcal{Q}}^{m-j}\mathcal{C}{\mathcal{Q}}^j\right){\mathcal{Q}}^n=0$. Our study systematically explores the algebraic properties and structural characterization of n-quasi-$[m,\mathcal{C}]$-symmetric operators through matrix representations, providing a deeper understanding of their internal structure. Moreover, we establish sufficient conditions under which the powers and products of such operators inherit the n-quasi-$[m,\mathcal{C}]$-symmetric property. Additionally, we investigate the tensor products of n-quasi-$[m,\mathcal{C}]$-symmetric operators. Finally, we identify conditions that distinguish n-quasi-$[m,\mathcal{C}]$-symmetric operators from n-quasi-$[m-1;\mathcal{C}]$-symmetric operators. Full article

The mathematical universe of the quantum topos, which is formulated on the basis of classical Boolean snapshots, delivers a neo-realist description of quantum mechanics that preserves realism. The main contribution of this article is developing formal objectivity in physical theories beyond quantum mechanics in the topos-theory approach. It will be shown that neo-realist responses to non-perturbative structures of quantum field theory do not preserve realism. In this regard, the method of Feynman graphons is applied to reframe the task of describing objectivity in quantum field theory in terms of replacing the standard Hilbert-space/operator-algebra ontology with a new context category built from a certain family of topological Hopf subalgebras of the topological Hopf algebra of renormalization as algebraic/combinatorial data tied to non-perturbative structures. This topological-Hopf-algebra ontology, which is independent of instrumentalist probabilities, enables us to reconstruct gauge field theories on the basis of the mathematical universe of the non-perturbative topos. The non-Boolean logic of the non-perturbative topos cannot be recovered by classical Boolean snapshots, which is in contrast to the quantum-topos reformulation of quantum mechanics. The article formulates a universal version of the non-perturbative topos to show that quantum field theory is a globally and locally neo-realist theory which can be reconstructed independent of the standard Hilbert-space/operator-algebra ontology. Formal objectivity of the universal non-perturbative topos offers a new route to build objective semantics for non-perturbative structures. Full article

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle ${\mathbb{S}}^1$, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras $\mathfrak{g}\left(\mathcal{M}\right)$, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on $SU\left(2\right)$, $SU\left(2\right)/U\left(1\right)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications. Full article

Using the Dixmier angle between two closed subspaces of a complex Hilbert space $\mathcal{H}$, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections, $P_{{\mathcal{W}}_1}$ and $P_{{\mathcal{W}}_2}$, onto closed subspaces ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$, to attain its maximum, namely $\parallel P_{{\mathcal{W}}_1}+P_{{\mathcal{W}}_2}\parallel =2.$ These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in $\mathcal{H}$, we derive sufficient conditions ensuring $\parallel T{T}^{†}+T^{†}T\parallel =2,$ where $T^{†}$ denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting. Full article

This paper extends the previous work by the author on m-null pairs of operators in Hilbert space. If an elementary operator L has elementary symbols A and B that are p-null and q-null, respectively, then L is $(p+q-1)$-null. Here, we prove the converse under strictness conditions, modulo some nonzero multiplicative constant—if L is strictly $(p+q-1)$-null, then a scalar $\lambda \ne 0$ exists such that $\lambda A$ is strictly p-null and ${\lambda }^{-1}B$ is strictly q-null. Our constructive argument relies essentially on algebraic and combinatorial methods. Thus, the result obtained by Gu on m-isometries is recovered without resorting to spectral analysis. For several operator classes that generalize m-isometries and are subsumed by m-null operators, the result is new. Full article

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

From an arbitrary definition of operators inspired by oscillators of Virasoro, an algebra is derived. It fits the structure of Virasoro algebra with null central charge or Witt algebra. The resulting formalism has yielded commutators with a dependence on integer numbers, and it follows the Witt-like algebra. Also, the quantum mechanics evolution operator for the case of the quantum harmonic oscillator was identified. Furthermore, the Schrödinger equation was systematically derived under the present framework. When operators are expressed in the framework of Hilbert space states, the resulting Witt algebra seems to be proportional to the well-known canonical commutation relation. This has demanded the development of a formalism based on arbitrary and physical operators as well as well-defined rules of commutation. The Witt-like was also redefined through the direct usage of the uncertainty principle. The results of the paper might suggest that Witt algebra encloses not only quantum mechanics’ fundamental commutator but also other unexplored relations among quantum mechanics observables and Witt algebra. Full article

In this paper, a novel class of meromorphic functions associated with the Mittag–Leffler function $E_{\mu ,\vartheta }\left(z\right)$ is introduced using the Hilbert space operator. In the punctured symmetric domain ${⋓}^{\ast }$, essential properties of this class are systematically investigated. These properties include coefficient inequalities, growth and distortion bounds, as well as weighted and arithmetic mean estimates. Furthermore, the extreme points and radii of geometric properties such as close-to-convexity, starlikeness, and convexity are analyzed in detail. Additionally, the Hadamard product (or convolution) is explored to demonstrate the algebraic structure and stability of the introduced function class under this operation. Integral mean inequalities are also established to provide further insights into the behavior of these functions within the given domain. Full article

In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method. Full article

Min and max matrices are structured matrices that appear in diverse mathematical and computational applications. Their inherent structures facilitate highly accurate numerical solutions to algebraic problems. In this research, the total positivity of generalized Min and Max matrices is characterized, and their bidiagonal factorizations are derived. It is also demonstrated that these decompositions can be computed with high relative accuracy (HRA), enabling the precise computations of eigenvalues and singular values and the solution of linear systems. Notably, the discussed approach achieves relative errors on the order of the unit roundoff, even for large and ill-conditioned matrices. To illustrate the exceptional accuracy of this method, numerical experiments on quantum extensions of Min and L-Hilbert matrices are presented, showcasing their superior precisions compared to those of standard computational techniques. Full article

This survey provides a review of the theoretical research on the classic system of matrix equations $AX=C$ and $XB=D$, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert $C^{\ast }$-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in color images. Full article

The Banach Gelfand Triple $(S_0,L^2,S_0^{\prime })$ consists of the Feichtinger algebra $S_0\left(R^d\right)$ as a space of test functions, the dual space $S_0^{\prime }\left(R^d\right)$, known as the space of mild distributions, and the intermediate Hilbert space $L^2\left(R^d\right)$. This Gelfand Triple is very useful for the description of mathematical problems in the area of time-frequency analysis, but also for classical Fourier analysis and engineering applications. Because the involved spaces are Banach spaces, we speak of a Banach Gelfand Triple, in contrast to the widespread concept of rigged Hilbert spaces, which usually involve nuclear Frechet spaces. Still, both concepts serve very similar purposes. Based on the manifold properties of $S_0\left(R^d\right)$, it has found applications in the derivation of mathematical statements related to Gabor Analysis but also in providing an alternative and more lucid description of classical results, such as the Shannon sampling theory, with a potential to renew the way how Fourier and time-frequency analysis, but also signal processing courses for engineers (or physicists and mathematicians) could be taught in the future. In the present study, we will demonstrate that one could choose a relatively large variety of similar Banach Gelfand Triples, even if one wants to include key properties such as Fourier invariance (an extended version of Plancherel’s Theorem). Some of them appeared naturally in the literature. It turns out, that $S_0\left(R^d\right)$ is the smallest member of this family. Consequently $S_0^{\prime }\left(R^d\right)$ is the largest dual space among all these spaces, which may be one of the reasons for its universal usefulness. This article provides a study of the basic properties following from a short list of relatively simple assumptions and gives a list of non-trivial examples satisfying these basic axioms. Full article

Let $\mathcal{A}$ be a self-adjoint standard operator algebra on a real or complex Hilbert space of dimension $\ge 2$, and let $k\in \{1,2,3\}$. The k-skew commutator for $A,B\in \mathcal{A}$ is defined by ${}_{\ast }[A,B]_1=AB-B{A}^{\ast }$ and ${}_{\ast }[A,B]_k={}_{\ast }[A,{}_1[A,{}_{\ast }[A,B]_{k-1}]_1$. Assume that $\mathsf{\Phi }:\mathcal{A}\to \mathcal{A}$ is a map whose range contains all rank-one projections. In this paper, we prove that $\mathsf{\Phi }$ is strong k-skew-commutativity preserving, that is, ${}_{\ast }[\mathsf{\Phi }\left(A\right),\mathsf{\Phi }\left(B\right)]_k={}_{\ast }[A,B]_k$ for all $A,B\in \mathcal{A}$ if and only if one of the following statements holds: (i) $\mathsf{\Phi }$ is either the identity map or the negative identity map whenever $k\in \{1,3\}$; (ii) $\mathsf{\Phi }$ is the identity map whenever $k=2$. Full article

Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well as an abstract basis and the basis of special functions and representations of Lie algebras of symmetries are used by continuous operators. This is not possible in Hilbert spaces. In the present paper, we study a model showing all these features, based on the one-dimensional Pöschl–Teller Hamiltonian. Also, RHS supports representations of all kinds of ladder operators as continuous mappings. We give an interesting example based on one-dimensional Hamiltonians with an infinite chain of SUSY partners, in which the factorization of Hamiltonians by continuous operators on RHS plays a crucial role. Full article