scholarly journals Further results on Left and Right Generalized Drazin Invertible Operators

2020 ◽  
Vol 54 (1) ◽  
pp. 98-106
Author(s):  
So. Messirdi ◽  
Sa. Messirdi ◽  
B. Messirdi
Keyword(s):  
Banach Space ◽  
Explicit Formula ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Left And Right ◽  
The Right ◽  
Generalized Drazin Invertible Operators ◽  
Jacobson’S Lemma

In this paper we present some new characteristics and expressions of left and right generalized Drazin invertible bounded operators on a Banach space $X.$ An explicit formula relating the left and the right generalized Drazin inverses to spectral idempotents is provided. In addition, we give a characterization of operators in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) with equal spectral idempotents, where $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) denotes the set of all left (resp. right) generalized Drazin invertible bounded operators on $X.$ Next, we give some sufficient conditions which ensure that the product of elements of $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) remains in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$). Finally, we extend Jacobson's lemma for left and right generalized Drazin invertibility. The provided results extend certain earlier works given in the literature.

On Fredholm properties of Lu = u′ − A(t)u for paths of sectorial operators

2005 ◽  
Vol 135 (1) ◽  
pp. 39-60 ◽  
Author(s):  
Davide di Giorgio ◽  
Alessandra Lunardi
Keyword(s):  
Banach Space ◽  
Explicit Formula ◽  
Fredholm Operator ◽  
Finite Dimension ◽  
Sufficient Conditions ◽  
Spectral Flow ◽  
Exponential Dichotomies ◽  
Sectorial Operators ◽  
Common Domain ◽  
General Banach Space

We consider a path of sectorial operators t ↦ A (t) ∈ Cα (R, L (D, X)), 0 < α < 1, in general Banach space X, with common domain D (A (t)) = D and with hyperbolic limits at ±∞. We prove that there exist exponential dichotomies in the half-lines (−∞, −T] and [T, +∞) for large T, and we study the operator (Lu)(t) = u′(t) − A(t)u(t) in the space Cα (R, D) ∩ C1+α (R, X). In particular, we give sufficient conditions in order that L is a Fredholm operator. In this case, the index of L is given by an explicit formula, which coincides to the well-known spectral flow formula in finite dimension. Such sufficient conditions are satisfied, for instance, if the embedding D ↪ X is compact.

scholarly journals Complex extreme measurable selections

1995 ◽  
Vol 58 (2) ◽  
pp. 222-231
Author(s):  
Douglas Mupasiri
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Measure Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Closed Unit Ball ◽  
Measurable Selections ◽  
Necessary And Sufficient

AbstractWe give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

Algebraic characterization of infinite Markov chains where movement to the right is limited to one step

1977 ◽  
Vol 14 (04) ◽  
pp. 740-747 ◽  
Author(s):  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Algebraic Characterization ◽  
Transient States ◽  
One Step ◽  
The Right

We consider an infinite Markov chain with states E 0, E 1, …, such that E 1, E 2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E 0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

scholarly journals Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1134 ◽  
Author(s):  
Mouffak Benchohra ◽  
Noreddine Rezoug ◽  
Bessem Samet ◽  
Yong Zhou
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Measure Of Noncompactness ◽  
Sufficient Conditions ◽  
Second Order ◽  
Fixed Point Method ◽  
Bounded Operators ◽  
Volterra Type ◽  
Kuratowski Measure Of Noncompactness ◽  
The Family

We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results.

Algebraic characterization of infinite Markov chains where movement to the right is limited to one step

1977 ◽  
Vol 14 (4) ◽  
pp. 740-747 ◽  
Author(s):  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Algebraic Characterization ◽  
Transient States ◽  
One Step ◽  
The Right

We consider an infinite Markov chain with states E0, E1, …, such that E1, E2, … is not closed, and for i ≧ 1 movement to the right is limited by one step. Simple algebraic characterizations are given for persistency of all states, and, if E0 is absorbing, simple expressions are given for the probabilities of staying forever among the transient states. Examples are furnished, and simple necessary conditions and sufficient conditions for the above characterizations are given.

scholarly journals Lateralization of face processing in the human brain

2012 ◽  
Vol 279 (1735) ◽  
pp. 2052-2061 ◽  
Author(s):  
Ming Meng ◽  
Tharian Cherian ◽  
Gaurav Singal ◽  
Pawan Sinha
Keyword(s):  
Face Processing ◽  
Inhibitory Influence ◽  
Hemispheric Differences ◽  
Response Dynamics ◽  
Activation Patterns ◽  
Face Selectivity ◽  
Left And Right ◽  
The Right ◽  
Processing Mechanisms

Are visual face processing mechanisms the same in the left and right cerebral hemispheres? The possibility of such ‘duplicated processing’ seems puzzling in terms of neural resource usage, and we currently lack a precise characterization of the lateral differences in face processing. To address this need, we have undertaken a three-pronged approach. Using functional magnetic resonance imaging, we assessed cortical sensitivity to facial semblance, the modulatory effects of context and temporal response dynamics. Results on all three fronts revealed systematic hemispheric differences. We found that: (i) activation patterns in the left fusiform gyrus correlate with image-level face-semblance, while those in the right correlate with categorical face/non-face judgements. (ii) Context exerts significant excitatory/inhibitory influence in the left, but has limited effect on the right. (iii) Face-selectivity persists in the right even after activity on the left has returned to baseline. These results provide important clues regarding the functional architecture of face processing, suggesting that the left hemisphere is involved in processing ‘low-level’ face semblance, and perhaps is a precursor to categorical ‘deep’ analyses on the right.

Sequent-systems for modal logic

1985 ◽  
Vol 50 (1) ◽  
pp. 149-168 ◽  
Author(s):  
Kosta Došen
Keyword(s):  
Order Logic ◽  
Philosophical Logic ◽  
Logical Constants ◽  
First Order ◽  
Sequent Systems ◽  
Left And Right ◽  
Level 1 ◽  
The Right ◽  
Level 2

AbstractThe purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else).This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1.These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical.

scholarly journals Localization theorems for matrices and bounds for the zeros of polynomials over quaternion division algebra

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 553-573 ◽  
Author(s):  
Sk. Ahmad ◽  
Istkhar Ali
Keyword(s):  
Zeros Of Polynomials ◽  
Sufficient Condition ◽  
Complex Matrix ◽  
Quaternionic Matrix ◽  
Left And Right ◽  
The Stability ◽  
The Right ◽  
Left And Right Eigenvalues ◽  
Quaternionic Polynomials

In this paper, we derive Ostrowski and Brauer type theorems for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic matrix. After that, a sufficient condition for the stability of a quaternionic matrix is given that generalizes the stability condition for a complex matrix. Finally, a characterization of bounds is derived for the zeros of quaternionic polynomials.

On the Operator Identity ∑ AkXBk ≡ 0

1978 ◽  
Vol 31 (4) ◽  
pp. 845-857 ◽  
Author(s):  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Operator Identity ◽  
Image Position ◽  
Space Case ◽  
Necessary And Sufficient ◽  
Induced Mapping ◽  
Compact Map

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by(1)We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the formFor example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

Network Architecture and the Left-Right Spectrum

2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Dmitry Taubinsky
Keyword(s):  
Network Architecture ◽  
Sufficient Conditions ◽  
Opinion Formation ◽  
Flip Flop ◽  
Long Run ◽  
Left And Right ◽  
Simple Sufficient Condition ◽  
Necessary And Sufficient ◽  
Over Time

We study a model of opinion formation and analyze the link between network architecture and the “left-right spectrum” that frequently characterizes opinions and beliefs. We correct a key result of DeMarzo, Vayanos and Zwiebel (QJE, 2003) who claim that after some time, an agent’s position on a set of different issues will always be either “left” on all of those issues or “right” on all of those issues. We provide counterexamples to this claim and show that in the long-run an agent’s position can flip-flop between “left” on all issues and “right” on all issues indefinitely. However, we provide necessary and sufficient conditions for a stable left-right characterization of opinions to be possible in the long run. Roughly, a flip-flop will occur when agents give relatively little weight to the opinions of agents with similar political positions (including themselves). Following this intuition, we show that a simple sufficient condition is that agents become “stubborn” over time and give little weight to the opinions of others. Finally, we characterize classes of networks in which it is possible for agents to flip-flop between “left” and “right” indefinitely. We argue that qualitatively, these results are robust to alternative models of opinion formation.

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