scholarly journals SPECTRAL CONDITIONS FOR UNIFORM P-ERGODICITIES OF MARKOV OPERATORS ON ABSTRACT STATES SPACES

2020 ◽  
pp. 1-15
Author(s):  
NAZIFE ERKURŞUN-ÖZCAN ◽  
FARRUKH MUKHAMEDOV
Keyword(s):  
Banach Space ◽  
Rate Of Convergence ◽  
Spectral Radius ◽  
Markov Operator ◽  
Asymptotical Stability ◽  
Ordered Banach Space ◽  
State Spaces ◽  
Markov Operators ◽  
Positive Elements

Abstract In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e. $\|T^n-P\|\to 0$ , here P is a projection. We have showed that T is uniformly P-ergodic if and only if $\|T^n-P\|\leq C\beta^n$ , $0<\beta<1$ . In this paper, we prove that such a β is characterized by the spectral radius of T − P. Moreover, we give Deoblin’s kind of conditions for the uniform P-ergodicity of Markov operators.

scholarly journals Approximations of non-homogeneous Markov chains on abstract states spaces

2021 ◽  
pp. 1-20
Author(s):  
Farrukh Mukhamedov ◽  
Ahmed Al-Rawashdeh
Keyword(s):  
Banach Space ◽  
Data Analysis ◽  
Markov Chains ◽  
Essential Role ◽  
Random Variables ◽  
Ordered Banach Space ◽  
State Spaces ◽  
Rank One ◽  
Probability And Statistics ◽  
Insight Into

Approximations of nonhomogeneous discrete Markov chains (NDMC) play an essential role in both probability and statistics. In all these settings, it is crucial to consider random variables in appropriate spaces. Therefore, the abstract considerations of such spaces lead to investigating the approximations in ordered Banach space scheme. In this paper, we consider two topologies on the set of NDMC of abstract state spaces. We establish that the set of all uniformly [Formula: see text]-ergodic NDMC is norm residual in NDMC. The set of point-wise weak [Formula: see text]-ergodic NDMC is also considered and such sets are shown to be a [Formula: see text]-subset (in strong topology) of NDMC. We point out that all the deduced results are new in the classical and non-commutative probabilities, respectively, since in most of earlier results the limiting projection is taken as a rank one projection. Indeed, the obtained results give new insight into data-analysis and statistics.

scholarly journals Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

2020 ◽  
Vol 18 (1) ◽  
pp. 858-872
Author(s):  
Imed Kedim ◽  
Maher Berzig ◽  
Ahdi Noomen Ajmi
Keyword(s):  
Banach Space ◽  
Positive Solution ◽  
Positive Definite ◽  
Positive Cone ◽  
Matrix Equations ◽  
Ordered Banach Space ◽  
Coincidence Points ◽  
Nonlinear Operators ◽  
Nonlinear Matrix

Abstract Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

Factorization of an adjontable Markov operator

2019 ◽  
Vol 22 (02) ◽  
pp. 1950013
Author(s):  
Carlo Pandiscia
Keyword(s):  
Hilbert Space ◽  
Matrix Algebra ◽  
Unitary Operator ◽  
Markov Operator ◽  
Factorization Property ◽  
Markov Operators ◽  
Dilation Theory ◽  
Probability Spaces

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

scholarly journals The local spectral radius of a nonnegative orbit of compact linear operators

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Mihály Pituk
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Additional Assumption ◽  
Real Banach Space ◽  
Linear Operators ◽  
Partial Ordering ◽  
Positive Eigenvector ◽  
Local Spectral Radius

AbstractWe consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

scholarly journals Weak Markov operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5453-5457
Author(s):  
Hūlya Duru ◽  
Serkan Ilter
Keyword(s):  
Positive Operator ◽  
Extreme Points ◽  
Markov Operator ◽  
Weak Order ◽  
Markov Operators

Let A and B be f -algebras with unit elements eA and eB respectively. A positive operator T from A to B satisfying T(eA) = eB is called a Markov operator. In this definition we replace unit elements with weak order units and, in this case, call T to be a weak Markov operator. In this paper, we characterize extreme points of the weak Markov operators.

scholarly journals Monotonic norms in ordered Banach spaces

1988 ◽  
Vol 45 (2) ◽  
pp. 217-219 ◽  
Author(s):  
K. F. Ng ◽  
C. K. Law
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Alternative Proof ◽  
Ordered Banach Space ◽  
Dual Result ◽  
Ordered Banach Spaces

AbstractLet B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

scholarly journals A certain fixed point theorem and its applications to integral-functional equations

1992 ◽  
Vol 46 (2) ◽  
pp. 179-186 ◽  
Author(s):  
M. Zima
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Linear Operator ◽  
Spectral Radius ◽  
Bounded Linear Operator ◽  
Functional Equations ◽  
Integral Functional ◽  
Bounded Linear ◽  
Banach’S Contraction Principle

In this paper a variant of Banach's contraction principle is established. By using the properties of the spectral radius of a bounded linear operator A defined in a suitable Banach space, we conclude that another operator A has exactly one fixed point in this space. In the second part of this paper some applications are given.

scholarly journals Absolute values in orthogonally decomposable spaces

1985 ◽  
Vol 31 (2) ◽  
pp. 215-233 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Ordered Banach Space ◽  
Locality Condition ◽  
Absolute Value ◽  
General Properties ◽  
The Absolute ◽  
Kato’S Inequality

In an ordered Banach space which is orthogonally decomposable, we define the absolute value and its general properties are given. The results are used to study the properties of linear operators which satisfy Kato's inequality and the locality condition.

Design of Optimal Fractional Luenberger Observers for Linear Fractional-Order Systems

2017 ◽  
Author(s):  
Arman Dabiri ◽  
Eric A. Butcher
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Fractional Order ◽  
State Transition ◽  
Design Method ◽  
Design Methods ◽  
The State ◽  
Transition Operator ◽  
Fractional Order Systems ◽  
Numerical Example

Optimal fractional Luenberger observers for linear fractional-order systems are developed using the fractional Chebyshev collocation (FCC) method. It is shown that the design method has advantages over existing Luenberger design methods for fractional order systems. To accomplish this, the state transition operator for the solution of linear fractional-order systems is defined in a Banach space and discretized using the FCC method. In addition, the discretized state transition operator is obtained by using the FCC method. Next, the optimal observer gains are obtained by minimizing the spectral radius of the state transition operator for the observer,while ensuring that the observer responds faster than the controller. Finally, a numerical example is provided to demonstrate the validity and the efficiency of the proposed method.

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