New Class Of Operators Research Articles

Polynomially accretive operators

In this paper, we introduce a new class of operators on a complex Hilbert space $\mathcal{H}$ which is called polynomially accretive operators, and thereby extending the notion of accretive and $n$--real power positive operators. We give several properties of the newly introduced class, and generalize some results for accretive operators. We also prove that every $2$--normal and $(2k+1)$--real power positive operator, for some $k\in\mathbb{N}$, must be $n$--normal for all $n\geq2$. Finally, we give sufficient conditions for the normality in the preceding implication.

Some Properties of (M,k)-quasi Paranormal Operators on Hilbert Spaces

Let H be a complex Hilbert space and let T represent a bounded linear operator on H. In this paper we introduce, a new class of non-normal operators, the (M,k)-quasi paranormal operator. An operator T is said to be (M,k)-quasi paranormal operator, for a non-negative integer k and a real positive number M, if it satisfies: ||Tk+1x||2 < M ||Tk+2x|| ||Tkx||, for every x in H. This new class of operators is a generalization of some of the non-normal operators, such as the k-quasi paranormal and M-paranormal operators. We prove the basic properties, the structural and spectral properties and also the matrix representation of this new class of operators.

Application of Improved Class of Weighted Averaging Operators in Intuitionistic Fuzzy Ann Guided MAGDM E-Commerce Problems

Objectives: To propose a novel and improvised Artificial Neural Network (ANN) technique to solve Multiple Attribute Group Decision Making (MAGDM) problems and an improved class of Aggregation operators for combining Intuitionistic Fuzzy Set (IFS) matrices for the ANN algorithm. Methods: A novel class of improved aggregation operators namely the Improved Intuitionistic Fuzzy Weighted Arithmetic Averaging (IM-IFWAA) operator and the Improved Intuitionistic Fuzzy Ordered Weighted Averaging (IM-IFOWA) operator are proposed in this work. The proposed improved class of aggregation operators will aggregate the IFS matrix data sets appearing in the form of matrices and then the revised input vectors which is then fed into the ANN algorithm following Delta, Perceptron, and Hebb Learning Rule for the next phase. Findings: Aggregating the Intuitionistic Fuzzy set information or data in today’s digital world is a tedious task in the field of MAGDM problem-solving. In this work, two new classes of operators for aggregating the data sets are proposed namely the IM-IFWAA operator and the IM-IFOWA operator which will improve the performance of the ANN. Necessary theorems for the proposed operators are proved to provide consistency of the same. The input created using the aforementioned operators is then fed into the novel ANN algorithm for further computations. Varieties of learning rules are then engaged for ranking of the best alternative for the decision problem. The developed theory is supported by a numerical example which is computed using all the proposed techniques. Novelty: Most of the research done on Intuitionistic Fuzzy Artificial Neural Network models are based on learning rules or using some other calculations. The proposed methods of using novel and improvised aggregation operators for ANN are used to find the inputs for ANN where varieties of learning rules for ANN are employed for effective decision analysis. Keywords: ANN, Learning Rules, MAGDM, Intuitionistic Fuzzy Sets, Weighted Aggregation Operator

On A-normaloid d-tuples of operators and related questions

Let ß A (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T 1 ,…, T d ) ∈ ß A (ℍ) d is called jointly A-normaloid if rA (T) = ∥T∥ A , where rA (T) and ∥T∥ A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple . Here . Finally, several related questions are explored.

B-AM-Dunford–Pettis Operators on Banach lattices

In our research work, we introduce a new class of operators that we call b-AM-Dunford–Pettis operators. Properties of b-AM-Dunford–Pettis operators, the relationship between the b-AM-Dunford–Pettis operators and various classes of operators are investigated. On the other side, our techniques and results will be related to the lattice structure of the b-AM-Dunford–Pettis operators. For instance, it will be proved that under certain conditions, the b-AM-Dunford–Pettis opertors verify the domination properties.

Approximation by modified (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-gamma-type operators

The main object of this paper is to construct a new class of modified (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven.

The Multi-Index Mittag-Leffler-Le Roy Functions as I- and H¯-Functions and New Fractional Calculus Operators

In the last decade, the interest in a special function named after the French mathematician Le Roy has provoked several authors to introduce and study its various extensions. Historically, the Le Roy function appeared almost in same time and in similar way of goals like the Mittag-Leffler function that has important role in Fractional Calculus. The Le Roy type functions are also “fractional exponentials” but the fractionalizing parameters appear as power indices of the involved Gamma functions. In our recent works, we have introduced rather general multi-index Le Roy type functions, studied their analytical properties and proposed their association to the class of Special Functions of Fractional Calculus.A newly developed idea is to show that the Le Roy type functions can be represented in terms of the I-functions of Rathie and H-functions of Inayat-Hussain that are further extensions of the Fox H¯-functions and Fox-Wright pΨq-functions. It happens that other important mathematical functions as the polylogarithms, Riemann Zeta function and its extensions, also belong to this more general class of special functions. We have solved, at least in some simpler case, the problem to find integral and differential-like operators for which the Le Roy type functions are eigenfunctions. This lead us to a new class of operators with I-functions kernels that can be considered as further extensions of the operators of generalized fractional calculus.

New extension of quasi-$ M $-hypnormal operators

This study introduces a new class of operators called polynomilally quasi-$ M $-hyponormal, which combining $ M $-hyponormal, quasi-$ M $-hyponormal, and $ k $-quasi-$ M $-hyponormal operators. We will demonstrate several properties of this class that correspond to those of $ k $-quasi-$ M $-hyponormal operators.

Approximation properties by Bézier variant of the Baskakov–Schurer–Szász–Stancu operators

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness.

Applications to Solving Variational Inequality Problems via MR-Kannan Type Interpolative Contractions

The aim of this paper is manifold. We first define the new class of operators called MR-Kannan interpolative type contractions, which includes the Kannan, enriched Kannan, interpolative Kannan type, and enriched interpolative Kannan type operators. Secondly, we prove the existence of a unique fixed point for this class of operators. Thirdly, we study Ulam-Hyers stability, well-posedness, and periodic point properties. Finally, an application of the main results to the variational inequality problem is given.

Classes of operators related to binormal operators

Binormal operators play a key part in the study of operator theory, particularly in testing spectral algorithms. A new class of operators known as the (n,m)-binormal operators is introduced in this paper, then we study its characteristics and relationship with other classes of operators. It is shown that if an operator T is (n,m)-binormal, and is unitarily quasiequivalent to an operator S, then S is also (n,m)-binormal.

N-Quasi-m-contractive and n-quasi-m-expansive operators on a Hilbert space

In this paper, we introduce a new class of operators called the [Formula: see text]-quasi-[Formula: see text]-contractive and [Formula: see text]-quasi-[Formula: see text]-expansive operators which are generalizations of [Formula: see text]-quasi-[Formula: see text]-isometric operators on Hilbert space introduced by Mecheri and Prasad, in [S. Mecheri and T. Prasad, On n-quasi-m-isometric operators, Asian-Eur. J. Math. 9(4) (2016) 1650073]. In addition, we apply those classes to the weighted composition operator. We discuss some examples concerning those classes and the weighted composition operator.

On the class of order almost L-weakly compact operators

We introduce a new class of operators that generalizes L-weakly compact operators, which we call order almost L-weakly compact. We give some characterizations of this class and we show that this class of operators satisfies the domination problem.

On the perturbation of pseudo-generalized invertible operators

This paper is a continuation of previous works Lahmar (Filomat 36:2551-2572, 2022), Lahmar (Filomat 36: 4575–4590, 2022), Lahmar (Preprint) where we defined a new class of operators called pseudo-generalized invertible operators that includes both the set of generalized invertible operators and the set of Drazin invertible operators. Here we focus essentially on the perturbation problem of pseudo-generalized invertible operators and the particular case of DPG invertibility.

Polynomially partial isometric operators

In order to extend the notion of semi-generalized partial isometries and partial isometries, we introduce a new class of operators called polynomially partial isometries. Since this new class of operators contains semi-generalized partial isometries, partial isometries, isometries and co-isometries, we proposed a wider class of operators. Several basic properties of polynomially partial isometries and some invariant subspaces of corresponding operators are presented. We study decomposition theorems and spectral theorems for polynomially partial isometries, generalizing some well-known results for partial isometries and semi-generalized partial isometries to polynomially partial isometries. Applying polynomially partial isometries, we solve some equations.

Quasi-normal Operator of order n

In this paper, we introduce a new class of operators acting on a complex Hilbert space H which is called quasi-normal operator of order n. An operator T∈B(H) is called quasi-normal operator of order n if T(T*n Tn)=(T*n Tn)T, where n is positive integer number greater than 1 and T* is the adjoint of the operator T, We investigate some basic properties of such operators and study relations among quasi-normal operator of order n and some other operators.

Classes of Operators Related to Polynomially Normal Operators

In this paper, we introduce new classes of operators related to the class of polynomially normal operators which are described as follows: (i) $$m$$ -quasi polynomially normal operators includes polynomially normal operators recently studied in [7, 6]. A bounded linear operator $$S$$ on a complex Hilbert space $$\mathcal{H}$$ is said to be $$m$$ -quasi polynomially normal operator if there exists a nontrivial polynomial $$P={\sum }_{0\le k\le n}{b}_{k}{z}^{k}\in {\mathbb{C}}[z]$$ for which $${S}^{*m}(P(S){S}^{*}-{S}^{*}P(S)){S}^{m}=\,0$$ $$(\Leftrightarrow \sum_{0\le k\le n}{b}_{k}{S}^{*m}({S}^{k}{S}^{*}-{S}^{*}{S}^{k}){S}^{m}=\,0),$$ where $$m$$ is a natural number. (ii) Polynomially $$C$$ -normal operators includes $$C$$ -normal operators studied in [13, 23, 25]. An operator $$S$$ is called polynomially $$C$$ -normal operator if there exists a nontrivial polynomial $$P\in {\mathbb{C}}[z]$$ and a conjugation operator $$C$$ on $$\mathcal{H}$$ for which $$CP(S){S}^{*}-{S}^{*}P(S)C=\,0(\Leftrightarrow \sum_{0\le k\le n}{b}_{k}(C{S}^{k}{S}^{*}-{S}^{*}{S}^{k}C)=\,0).$$ A detailed study of certain properties of some members of the first class has been presented. However, an initiation to the study of the second class has been given.

Iterative schemes for numerical reckoning of fixed points of new nonexpansive mappings with an application

<abstract><p>The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $ (\alpha, \beta, \gamma) $-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $ K $-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $ K $-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $ K $-type iterative method to solve a variational inequality problem on a Hilbert space.</p></abstract>

Anisotropic elliptic equations with gradient-dependent lower order terms and $ L^1 $ data

<abstract><p>We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $ \mathcal Au+\Phi(x, u, \nabla u) = \mathfrak{B}u+f $ in $ \Omega $, where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ and $ f\in L^1(\Omega) $ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $ \mathcal A $, the prototype of which is $ \mathcal A u = -\sum_{j = 1}^N \partial_j(|\partial_j u|^{p_j-2}\partial_j u) $ with $ p_j > 1 $ for all $ 1\leq j\leq N $ and $ \sum_{j = 1}^N (1/p_j) > 1 $. As a novelty in this paper, our lower order terms involve a new class of operators $ \mathfrak B $ such that $ \mathcal{A}-\mathfrak{B} $ is bounded, coercive and pseudo-monotone from $ W_0^{1, \overrightarrow{p}}(\Omega) $ into its dual, as well as a gradient-dependent nonlinearity $ \Phi $ with an "anisotropic natural growth" in the gradient and a good sign condition.</p></abstract>

Note on a new class of operators between some spaces of holomorphic functions

<abstract><p>The boundedness and compactness of a new class of linear operators from the weighted Bergman space to the weighted-type spaces on the unit ball are characterized.</p></abstract>