Properties of generalized strongly Drazin invertible elements in general rings

2017 ◽  
Vol 16 (11) ◽  
pp. 1750207 ◽  
Author(s):  
Orhan Gürgün
Keyword(s):  
General Setting ◽  
Drazin Inverse ◽  
Invertible Elements ◽  
General Ring

In this paper, we define the generalized strong Drazin inverse in a general ring and investigate this class of inverses. Thus, recent results on the strong Drazin invertible and generalized strong Drazin invertible elements are extended to a more general setting. In particular, we show that [Formula: see text] is generalized strong Drazin invertible in a general ring [Formula: see text] if and only if there exists an idempotent [Formula: see text] such that [Formula: see text] and [Formula: see text] is quasinilpotent in [Formula: see text]. We also prove that if [Formula: see text] is generalized Drazin invertible in [Formula: see text] for some [Formula: see text], so are [Formula: see text], [Formula: see text], [Formula: see text]. This partially answer to a question posed by Mosić.

Central Drazin inverses

2019 ◽  
Vol 18 (04) ◽  
pp. 1950065 ◽  
Author(s):  
Cang Wu ◽  
Liang Zhao
Keyword(s):  
Drazin Inverse ◽  
Reverse Order ◽  
Basic Properties ◽  
Reverse Order Law ◽  
Invertible Elements

We introduce and study a subclass of the Drazin invertible elements in a ring [Formula: see text], which are called central Drazin invertible. An element [Formula: see text] is said to be central Drazin invertible if there exists [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] for some integer [Formula: see text]. Some basic properties of the central Drazin inverse are obtained. Of particular interest are the central Drazin invertible elements that are simultaneously group invertible, which we show have a property generalizing strong cleanness. Some well-known results related to the cleanness of rings and the reverse order law are generalized.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

2019 ◽  
Vol 18 (11) ◽  
pp. 1950212 ◽  
Author(s):  
Honglin Zou ◽  
Dijana Mosić ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
Upper Bounds ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Operator Matrices ◽  
Skew Field ◽  
Block Matrices ◽  
Necessary And Sufficient ◽  
Drazin Index

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

scholarly journals The inverse along an element in rings

2016 ◽  
Vol 31 ◽  
pp. 572-592 ◽  
Author(s):  
Julio Benitez ◽  
Enrico Boasso
Keyword(s):  
Drazin Inverse ◽  
Group Inverse ◽  
Fixed Element ◽  
Special Cases ◽  
Generalized Drazin Inverse ◽  
Invertible Elements

Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

2019 ◽  
Vol 62 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Marcell Gaál
Keyword(s):  
Convex Combination ◽  
General Setting ◽  
Positive Semidefinite ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Density Matrices ◽  
Positive Semidefinite Matrices ◽  
Von Neumann ◽  
Convex Functionals ◽  
Invertible Elements

AbstractVery recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten $p$-norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a $C^{\ast }$-algebra.

scholarly journals On the weighted pseudo Drazin invertible elements in associative rings and Banach algebras

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6359-6367
Author(s):  
Jianlong Chen ◽  
Xiaofeng Chen ◽  
Hassane Zguitti
Keyword(s):  
Banach Algebras ◽  
Drazin Inverse ◽  
Invertible Elements ◽  
Associative Rings ◽  
Equivalent Conditions

In this paper, we introduce and investigate the weighted pseudo Drazin inverse of elements in associative rings and Banach algebras. Some equivalent conditions for the existence of the w-pseudo Drazin inverse of a + b are given. Using the Pierce decomposition, the representations for the w-pseudo Drazin inverse are given in Banach algebras.

On the Topologically Invertible Elements of a Topological Algebra

2007 ◽  
Vol 107 (1) ◽  
pp. 73-80
Author(s):  
Hugo Arizmendi-Peimbert ◽  
Angel Carrillo-Hoyo
Keyword(s):  
Topological Algebra ◽  
Invertible Elements

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

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