Self-adjoint Operators Research Articles

On self-adjoint operators generated by the fourth-order Laguerre type differential expression

In this paper we address the problem of finding all self-adjoint operators generated by a Lagrangian symmetrizable differential expression in a Hilbert space with a mixed Sobolev inner product strictly discrete on the derivatives. We start giving survey on the initial results in the literature: the Glazman-Krein-Naimark (GKN) theory, the later developed GKN-EM theory as an extension of the first, and the works of Littlejohn and Wellman on the subject.We apply those to the fourth-order Laguerre type differential expression known for having a sequence of orthogonal polynomials as eigenfunctions and a symmetrizable differential expression. As a result, we give a complete description of the operator space defined from the GKN-EM theory, and also a characterization of all possible self-adjoint operators in the Sobolev space where the operator is defined. We validate these results with known examples previously studied in the literature.

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

An extended COPRAS method based on complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean aggregation operators

An extended COPRAS method based on complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean aggregation operators

Multi-criteria Radio Frequency Identification Approach for Manufacturing Company Selection Based on Partitioned Maclaurin Symmetric Mean Operators Under Complex Intuitionistic Fuzzy Set Environment

Multi-criteria Radio Frequency Identification Approach for Manufacturing Company Selection Based on Partitioned Maclaurin Symmetric Mean Operators Under Complex Intuitionistic Fuzzy Set Environment

Analysis of a nonlinear model for the thermoelastic beam

This paper addresses the Cauchy problem associated with the nonlinear thermoelastic beam featuring thermal dissipation. We examine the problem in an open domain, which may be either bounded or unbounded. The existence of a solution is derived through the diagonalization theorem for self-adjoint operators using Hilbertian integral theory.

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

Differential and fuzzy differential sandwich theorems involving quantum calculus operators

The principle of subordination is useful in comparing two holomorphic functions when the range of one holomorphic function is a subset of the other and they comply at a single point. The subordination, when spoken in fuzzy set theory, becomes fuzzy subordination as the comparison between two holomorphic functions is made using the fuzzy membership function. In this article, differential and fuzzy differential Subordination, superordination, and sandwich theorems have been discussed for the classes defined by using q-derivative and symmetric q-derivative operators.

Browder operators on quaternionic Hilbert space

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators.

Refinements and Reverses of Tensorial and Hadamard Product Inequalities for Selfadjoint Operators in Hilbert Spaces Related to Young's Result

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $<$ $m\leq A,$ $B\leq M,$ for some constants $m,$ $M,$ then \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \\ & \leq \left( 1-\nu \right) A\otimes 1+\nu 1\otimes B-A^{1-\nu }\otimes B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}\otimes 1+1\otimes B^{2}}{2}-A\otimes B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$ We also have the inequalities for Hadamard product \begin{align*} 0& \leq \frac{m}{M^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \\ & \leq \left[ \left( 1-\nu \right) A+\nu B\right] \circ 1-A^{1-\nu }\circ B^{\nu } \\ & \leq \frac{M}{m^{2}}\nu \left( 1-\nu \right) \left( \frac{A^{2}+B^{2}}{2}% \circ 1-A\circ B\right) \end{align*} for all $\nu \in \left[ 0,1\right] .$

On Perturbation of Operators and Rayleigh-Schrödinger Coefficients

Let A and E be self-adjoint matrices or operators on ℓ2(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2({{\\mathbb {N}}})$$\\end{document}, where A is fixed and E is a small perturbation. We study how the eigenvalues of A+E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+E$$\\end{document} depend on E, with the aim of obtaining second order formulas that are explicitly computable in terms of the spectral decomposition of A and a certain block decomposition of E. In particular we extend the classical Rayleigh-Schrödinger formulas for the one-parameter perturbation A+tE\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A+tE$$\\end{document} where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in {{\\mathbb {R}}}$$\\end{document} varies and E is held fixed, by dropping t and considering E as the variable.

Stability of the gapless pure point spectrum of self-adjoint operators

We consider a self-adjoint operator T on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of T and on the bounded perturbation V ensuring the global stability of the spectral nature of T + ɛV, ε∈R.

Upper bounds for Berezin numbers of self-adjoint operators and applications

We prove upper bounds for Berezin symbols and Berezin numbers of operators, with a special emphasis on the quadratic weighted operator geometric mean. In particular, we prove some upper bounds for Berezin symbols and Berezin numbers of some self-adjoint operators on reproducing kernel Hilbert spaces over some suitable sets. Moreover, we estimate the best possible constant in some generalized Hardy-Hilbert inequality for the series.

The τ$$ \tau $$‐symmetries and Lie algebra structure of the Blaszak–Marciniak lattice equation

A general approach is developed for discriminating strong and hereditary symmetric operators. The recursion operator of the Blaszak–Marciniak (BM) equation hierarchy is proved to be strong and hereditary symmetric. As an example of discrete soliton equations related to matrix spectral problems, the ‐symmetries and Lie algebra structure of the BM equation are built firstly.

Quantum current and holographic categorical symmetry

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C}𝒞:=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C}𝒞 and symmetric operators are morphisms in \mathcal{C}𝒞. The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C})Z1(𝒞). The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C})Z1(𝒞) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

Solving Particle–Antiparticle and Cosmological Constant Problems

We solve the particle-antiparticle and cosmological constant problems proceeding from quantum theory, which postulates that: various states of the system under consideration are elements of a Hilbert space H with a positive definite metric; each physical quantity is defined by a self-adjoint operator in H; symmetry at the quantum level is defined by a representation of a real Lie algebra A in H such that the representation operator of any basis element of A is self-adjoint. These conditions guarantee the probabilistic interpretation of quantum theory. We explain that in the approaches to solving these problems that are described in the literature, not all of these conditions have been met. We argue that fundamental objects in particle theory are not elementary particles and antiparticles but objects described by irreducible representations (IRs) of the de Sitter (dS) algebra. One might ask why, then, experimental data give the impression that particles and antiparticles are fundamental and there are conserved additive quantum numbers (electric charge, baryon quantum number and others). The reason is that, at the present stage of the universe, the contraction parameter R from the dS to the Poincare algebra is very large and, in the formal limit R→∞, one IR of the dS algebra splits into two IRs of the Poincare algebra corresponding to a particle and its antiparticle with the same masses. The problem of why the quantities (c,ℏ,R) are as are does not arise because they are contraction parameters for transitions from more general Lie algebras to less general ones. Then the baryon asymmetry of the universe problem does not arise. At the present stage of the universe, the phenomenon of cosmological acceleration (PCA) is described without uncertainties as an inevitable kinematical consequence of quantum theory in semiclassical approximation. In particular, it is not necessary to involve dark energy the physical meaning of which is a mystery. In our approach, background space and its geometry are not used and R has nothing to do with the radius of dS space. In semiclassical approximation, the results for the PCA are the same as in General Relativity if Λ=3/R2, i.e., Λ>0 and there is no freedom for choosing the value of Λ.

Experimental Investigation of Uncertainty Relations for Non-Hermitian Operators.

Uncertainty relations for Hermitian operators have been confirmed through many experiments. However, previous experiments have only tested the special case of non-Hermitian operators, i.e., uncertainty relations for unitary operators. In this study, we explore uncertainty relations for general non-Hermitian operators, which include Hermitian and unitary operators as special cases. We perform experiments with both real and complex non-Hermitian operators for qubit states, and confirm the validity of the uncertainty relations within the experimental error. Our results provide experimental evidence of uncertainty relations for non-Hermitian operators. Furthermore, our methods for realizing and measuring non-Hermitian operators are valuable in characterizing open-system dynamics and enhancing parameter estimation.

The symbol approximation method: a numerical approach to the approximation of the symbol of self-adjoint operators

The symbol approximation method: a numerical approach to the approximation of the symbol of self-adjoint operators

Pointwise density estimation on metric spaces and applications in seismology

AbstractWe are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general Hölder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.

Properties and Applications of Symmetric Quantum Calculus

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals.

Topological slow-light in one-dimensional conjugated photonic systems

Topological slow-light in a one-dimensional conjugated topological photonic crystal (CTPhC) is presented. The study shows that extremely slow-light of 26.57 m/s group velocity and 7.52 × 10−7 s group delay are achieved in the proposed CTPhC with 20 μm thick, which is better than traditional approaches of slow-light. Results also show that there are two significant sensitive parameters for slow-light in the CTPhC. Robust and ultra-slow-light is achieved as the refractive index contrast of materials and the number of layers increases. It is worth noting that the CTPhC has a symmetric and wide operating range photonic bandgap, which has better slow-light effects than typical topological photonic crystal. It is expected that the CTPhC can provide an efficient approach to constructing robust and ultra-slow-light devices.