Normal Operators Research Articles

Certain properties involving the unbounded operators p(T), TT⁎, and T⁎T; and some applications to powers and nth roots of unbounded operators

In this paper, we are concerned with conditions under which [p(T)]⁎=p‾(T⁎), where p(z) is a one-variable complex polynomial, and T is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the identities σ(AB)=σ(BA), where A and B are two unbounded operators. The equations (TT⁎)⁎=TT⁎ and (T⁎T)⁎=T⁎T, where T is a densely defined closable operator, are also studied. A particular interest will be paid to the equation T⁎T=p(T) and its variants. Then, we have certain results concerning nth roots of classes of normal and nonnormal (unbounded) operators. Some further consequences and counterexamples accompany our results.

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.

Higher rank numerical ranges of normal operators and unitary dilations

Here we give a closure free description of the higher rank numerical range of a normal operator acting on a separable Hilbert space. This generalizes a result of Avendaño for self-adjoint operators. It has several interesting applications. We show using Durszt's example that there exists a normal contraction T for which the intersection of the higher rank numerical ranges of all unitary dilations of T contains the higher rank numerical range of T as a proper subset. We strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it. We also show that the above condition is necessary for a general contraction.

An omnidirectional seismic image extension

This paper introduces an operator, which extends the Born operator for the linearized variable density acoustic wave equation. Inspired by angle dependent scattering, I will define it as acting on distributions depending on space and angle coordinates. I will show that it can be written as a weighted integral of space-shift extended operators associated to arbitrary directions after a suitable coordinate transform. I will also formulate conditions under which its normal operator is a pseudo-differential operator. These will be seen to be less restrictive than the ones imposed on horizontal space-shift extended operators in the sense that they allow diving wave paths.

Cut-off kinks

We answer the question: If a vacuum sector Hamiltonian is regularized by an energy cutoff, how is the one-kink sector Hamiltonian regularized? We find that it is not regularized by an energy cutoff, indeed normal modes of all energies are present in the kink Hamiltonian, but rather the decomposition of the field into normal mode operators yields coefficients which lie on a constrained surface that forces them to become small for energies above the cutoff. This explains the old observation that an energy cutoff of the kink Hamiltonian leads to an incorrect one-loop kink mass. To arrive at our conclusion, we impose that the regularized kink sector Hamiltonian is unitarily equivalent to the regularized vacuum sector Hamiltonian. This condition implies that the two regularized Hamiltonians have the same spectrum and so guarantees that the kink Hamiltonian yields the correct kink mass.

Unitary Group Orbits Versus Groupoid Orbits of Normal Operators

We study the unitary orbit of a normal operator $$a\in {{\mathcal {B}}}({{\mathcal {H}}})$$ , regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the various ambient spaces if and only if the spectrum of a is finite, and in that case, it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of a is not finite, we describe the closure of the orbits of a for the different norm topologies involved. We relate these results to the action of the groupoid of partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator a, we also describe the norm closure of its groupoid orbit $${{\mathcal {O}}}_a$$ , which leads to necessary and sufficient spectral conditions on a ensuring that $${{\mathcal {O}}}_a$$ is norm closed and that $${{\mathcal {O}}}_a$$ is a closed embedded submanifold of $${{\mathcal {B}}}({{\mathcal {H}}})$$ .

ON THE CLASS OF $n$-NORMAL OPERATORS AND MOORE-PENROSE INVERSE

Let $ T \in B(H)$ be a bounded linear operator on a complex Hilbert space $H$. For $ n\in \mathbb{N } $, an operator $ T\in B(H)$ is said to be n-normal if $ T^{n}T^{*}=T^{*}T^{n} $. In this paper we investigate a necessary and sufficient condition for the n-normality of $ ST $ and $ TS $, where $ S,T \in B(H). $ As a consequence, we generalize Kaplansky theorem for normal operators to n-normal operators. Also, In this paper, we provide new characterizations of n-normal operators by certain conditions involving powers of Moore-Penrose inverse.

A~normal operator on a b-Hilbert space

In this paper, we introduce ~normal operator on a b-Hilbert space, we will discuss some properties of these operators. Study b-spectrum of ~normal operator on finite dimension b-Hilbert space, we show that b-eigenvectors of ~normal operator are b-orthogonal.

Fractional resolvent family generated by normal operators

<abstract><p>The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for $ \{S_{\alpha}(t)\} $ if $ \alpha \neq 1 $; and we show that if $ \{S_{\alpha}(t)\} $ is stable, then we can determine the decay rate by $ \sigma(A) $ and some examples are given; we also prove that $ S_{\alpha}(t)x $ has a continuous derivative of order $ \alpha\beta > 0 $ if and only if $ x \in D(I-A)^{\beta} $. The main method we used here is the resolution of identity corresponding to a normal operator $ A $ and spectral measure integral.</p></abstract>

Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein–Milman Theorem, we prove that a rank $1$ linear operator of unit norm between such spaces can be expressed as a convex combination o

The Conceptual Foundations of the Sociology of Law by Pitirim Sorokin and Max Weber

The socio-legal theories of P. A. Sorokin and M. Weber are common in belonging to the normativist trend in the sociology of law. To determine the limits of their applicability to the analysis of the operation of legal norms, it is necessary to clarify the reasons for their differences. The notable differences between both theories are connected not only with the general differences in corresponding sociological systems, but also with fundamentally different conceptual foundations of socio-legal doctrines. The Russian-American sociologist P. A.Sorokin, in developing the ideas of L. I. Petrażycki, considered law as a set of norms with a certain content which indicates the permitted and proper behavior by means of the distribution of rights and obligations, which are always thought to be inextricably linked. This understanding of legal norms allows us to meaningfully separate them from the norms of morality, etiquette, technical norms, and rules of fashion. A key feature of law as a special kind of legitimate order for Weber is its coercion, ensured by the staff, i.e., a group of people specifically aimed at forcing compliance with the order. In contrast to Sorokin, Weber believed that law and other related phenomena are distinguished not at the level of individual norms, but at the level of normative systems (orders). Sorokin focused on organized groups, the skeleton of which are the norms that both determine the behavior of group members and create its structure. Contrary to Sorokin, Weber believes that normative motivation is only able to influence human behavior, but not to determine it. The difference in the researchers’ perceptions of the importance of normative motivation may be related to the focus on active duties in the case of Sorokin and on the framework model of the behavior of the empowered person in Weber’s case.

Robot sensors process based on generalized Fermatean normal different aggregation operators framework

<abstract><p>Novel methods for multiple attribute decision-making problems are presented in this paper using Type-Ⅱ Fermatean normal numbers. Type-Ⅱ Fermatean fuzzy sets are developed by further generalizing Fermatean fuzzy sets and neutrosophic sets. The Type-Ⅱ Fermatean fuzzy sets with basic aggregation operators are constructed. The concept of a Type-Ⅱ Fermatean normal number is compatible with both commutative and associative rules. This article presents a new proposal for Type-Ⅱ Fermatean normal weighted averaging, Type-Ⅱ Fermatean normal weighted geometric averaging, Type-Ⅱ generalized Fermatean normal weighted averaging, and Type-Ⅱ generalized Fermatean normal weighted geometric averaging. Furthermore, these operators can be used to develop an algorithm that solves MADM problems. Applications for the Euclidean distance and Hamming distances are discussed. Finally, the sets that arise as a result of their connection to algebraic operations are emphasized in our discourse. Examples of real-world applications of enhanced Hamming distances are presented. A sensor robot's most important components are computer science and machine tool technology. Four factors can be used to evaluate the quality of a robotics system: resolution, sensitivity, error and environment. The best alternative can be determined by comparing expert opinions with the criteria. As a result, the proposed models' outcomes are more precise and closer to integer number $ \delta $. To demonstrate the applicability and validity of the models under consideration, several existing models are compared with the ones that have been proposed.</p></abstract>

Some Properties of Range Operators on LCA Groups

In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in L2(G), where G is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from L2(ℝd) to L2(G). We investigate the matrix structure of range operator R(ξ) on range function J associated to shift-invariant space V , in the case of a locally compact Abelian group G. We also focus on some properties like as normal and unitary operator for range operator on L2(G). We show that shift preserving operator U is invertible if and only if fiber of corresponding range operator R is invertible and investigate the measurability of inverse R−1(ξ) of range operator on L2(G).

A note on intuitionistic fuzzy-n-normal operator on IFH-space

In this article, we have defined a new type of Normal operator on IFH-space, proven some important properties in it, and proved some mathematical operations based on the definition of this type.

Spectral representation of absolutely minimum attaining unbounded normal operators

Spectral representation of absolutely minimum attaining unbounded normal operators

Aczel-Alsina-based aggregation operators for intuitionistic hesitant fuzzy set environment and their application to multiple attribute decision-making process

<abstract> <p>An intuitionistic hesitant fuzzy set is an extension of the fuzzy set which deals with uncertain information and vague environments. Multiple-attribute decision-making problems (MADM) are one of the emerging topics and an aggregation operator plays a vital role in the aggregate of different preferences to a single number. The Aczel-Alsina norm operations are significant terms that handle the impreciseness and undetermined data. In this paper, we build some novel aggregation operators for the different pairs of the intuitionistic hesitant fuzzy sets (IHFSs), namely as Aczel-Alsina average and geometric operators. Several characteristics of the proposed operators are also described in detail. Based on these operators, a multi-attribute decision-making algorithm is stated to solve the decision-making problems. A numerical example has been taken to display and validate the approach. A feasibility and comparative analysis with existing studies are performed to show its superiority.</p> </abstract>

The -isomorphism property for a class of singularly-weighted x-ray transforms

We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk , obtained by considering specific singularly weighted L 2 topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.

Stability estimates for the X-ray transform on simple asymptotically hyperbolic manifolds

We study the normal operator to the geodesic X-ray transform on functions in the setting of simple asymptotically hyperbolic manifolds. We construct a parametrix for the normal operator in the 0-pseudodifferential calculus and use it show a stability estimate.

Urbanizing Human Rights Law: Cities, Local Governance and Corporate Power

AbstractThis article considers ways in which human rights law ought to respond to a growingly urban global order of blurred private—corporate—and state power. Fragmented and dispersed power comes together, in different configurations of public and private, in the cities and towns of the world. For this reason, local government presents the appropriate scale at which to re-conceptualize the operation of international human rights norms, also against private power. This requires engaging not only with the reach and leanings of international human rights standards but also with the manner in which they are rendered applicable, through domestic constitutional law, against state and non-state actors at a local scale. The urbanization of human rights law accordingly also requires a second look at the powers, competencies and responsibilities of urban local government under domestic constitutional law.

Generalized Differentiation of a Class of Normal Cone Operators and Sensitivity of Optimal Control Problems

Generalized Differentiation of a Class of Normal Cone Operators and Sensitivity of Optimal Control Problems