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 On the mean convergence of Markov operators

1974 ◽  
Vol 19 (2) ◽  
pp. 205-209 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Positive Operator ◽  
Markov Operator ◽  
Probability Measures ◽  
Linear Functionals ◽  
Markov Operators ◽  
Mean Convergence ◽  
Unit Function ◽  
The Mean ◽  
Borel Probability ◽  
Image Position

Throughout the paper, T will be a Markov operator on C(X) (X compact T2), i.e. a continuous positive operator such that Te = e (e the unit function). P will be the set of Borel probability measures on X, which we shall often think of as linear functionals on C(X), and , where T' is the adjoint of T. Let

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 Entanglement Breaking Channels

2003 ◽  
Vol 15 (06) ◽  
pp. 629-641 ◽  
Author(s):  
Michael Horodecki ◽  
Peter W. Shor ◽  
Mary Beth Ruskai
Keyword(s):  
Density Matrix ◽  
Positive Operator ◽  
Extreme Points ◽  
Completely Positive ◽  
Classical Quantum ◽  
Positive Operator Valued Measure ◽  
Special Classes

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑kRk Tr Fkρ where each Rk is a density matrix and Fk>0. If, in addition, Φ is trace-preserving, the {Fk} must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.

Generalized Markov Projections and Matrix Summability

1979 ◽  
Vol 22 (3) ◽  
pp. 311-316 ◽  
Author(s):  
Robert E. Atalla
Keyword(s):  
Sufficient Conditions ◽  
Markov Operator ◽  
Necessary And Sufficient Conditions ◽  
Averaging Operators ◽  
Matrix Summability ◽  
Markov Operators ◽  
Main Application ◽  
Summability Methods ◽  
Inclusion Relations ◽  
Necessary And Sufficient

In [A1] is defined a class of Markov operators on C(X) (X compact T2), called Generalized Averaging Operators (g.a.o.) which yield an easy solution to the following problem: given a fixed Markov operator T, find necessary and sufficient conditions on any other Markov operator R for the relation ker T ⊂ker R to hold. The main application of this is to inclusion relations between matrix summability methods.

scholarly journals On typical Markov operators acting on Borel measures

2005 ◽  
Vol 2005 (5) ◽  
pp. 489-497 ◽  
Author(s):  
Tomasz Szarek
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Markov Operator ◽  
Baire Category ◽  
Asymptotically Stable ◽  
Markov Operators ◽  
Borel Measures

It is proved that, in the sense of Baire category, almost every Markov operator acting on Borel measures is asymptotically stable and the Hausdorff dimension of its invariant measure is equal to zero.

scholarly journals Algebraic polymorphisms

2008 ◽  
Vol 28 (2) ◽  
pp. 633-642 ◽  
Author(s):  
KLAUS SCHMIDT ◽  
ANATOLY VERSHIK
Keyword(s):  
Algebraic Geometry ◽  
Invariant Measure ◽  
Special Class ◽  
Markov Operator ◽  
Point Of View ◽  
Compact Groups ◽  
Rational Matrix ◽  
Conjugate Operator ◽  
Markov Operators ◽  
Toral Automorphisms

AbstractIn this paper we consider a special class of polymorphisms with invariant measure, the algebraic polymorphisms of compact groups. A general polymorphism is—by definition—a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e. a positive operator on L2 of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of a torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators. A toral polymorphism is an automorphism of $\mathbb {T}^m$ if and only if the associated rational matrix lies in $\mathrm {GL}(m,\mathbb {Z})$. We characterize toral polymorphisms which are factors of toral automorphisms.

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.

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

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