Uniform ergodicities and perturbation bounds of Markov chains on ordered Banach spaces
aa r X i v : . [ m a t h . F A ] M a y UNIFORM ERGODICITIES AND PERTURBATION BOUNDS OFMARKOV CHAINS ON ORDERED BANACH SPACES
NAZIFE ERKURS¸UN ¨OZCAN AND FARRUKH MUKHAMEDOV ∗ Abstract.
It is known that Dobrushin’s ergodicity coefficient is one of the effectivetools in the investigations of limiting behavior of Markov processes. Several interest-ing properties of the ergodicity coefficient of a positive mapping defined on orderedBanach space with a base have been studied. In this paper, we consider uniformlymean ergodic and asymptotically stable Markov operators on ordered Banach spaces.In terms of the ergodicity coefficient, we prove uniform mean ergodicity criterion interms of the ergodicity coefficient. Moreover, we develop the perturbation theory foruniformly asymptotically stable Markov chains on ordered Banach spaces. In partic-ularly, main results open new perspectives in the perturbation theory for quantumMarkov processes defined on von Neumann algebras. Moreover, by varying the Ba-nach spaces one can obtain several interesting results in both classical and quantumsettings as well. Introduction
It is well-known that the transition probabilities P ( x, A ) (defined on a measur-able space ( E, F )) of Markov processes naturally define a linear operator by T f ( x ) = R f ( y ) P ( x, dy ), which is called Markov operator and acts on L -spaces. The study ofthe entire process can be reduced to the study of the limit behavior of the correspondingMarkov operator (see [12]). When we look at quantum analogous of Markov processes,which naturally appear in various directions of quantum physics such as quantum sta-tistical physics and quantum optics etc. In these studies it is important to elaboratewith associated quantum dynamical systems (time evolutions of the system) [18], whicheventually converge to a set of stationary states. From the mathematical point of view,ergodic properties of quantum Markov operators were investigated by many authors.We refer a reader to [1, 7, 17] for further details relative to some differences betweenthe classical and the quantum situations.In [19] it was proposed to investigate ergodic properties of Markov operator on ab-stract framework, i.e. on ordered Banach spaces. Since the study of several propertiesof physical and probabilistic processes in abstract framework is convenient and impor-tant (see [2]). Some applications of this scheme in quantum information have beendiscussed in [18]. We emphasize that the classical and quantum cases confine to thisscheme. We point out that in this abstract scheme one considers an ordered normedspaces and mappings of these spaces (see [2]). Moreover, in this setting mostly, certainergodic properties of Markov operators s were considered and investigated in [3, 6, 18].Nevertheless, the question about the sensitivity of stationary states and perturbations Date : Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author.2010
Mathematics Subject Classification.
Primary 47A35; Secondary 60J10, 28D05.
Key words and phrases. uniformly asymptotically stable, uniformly mean ergodic, Markov operator,norm ordered space, Dobrushin’s coefficient, perturbation bound. of the Markov chain are not explored well. Very recently in [21], perturbation boundshave been found for a quantum Markov chains acting on finite dimensional algebras.On the other hand, it is known [11, 13] that Dobrushin’s ergodicity coefficient isone of the effective tools in the investigations of limiting behavior of Markov processes(see [10, 20] for review). In [15, 16] we have defined such an ergodicity coefficient δ ( T ) of a positive mapping T defined on ordered Banach space with a base, and stud-ied its properties. In this paper, we consider uniformly mean ergodic and uniformlyasymptotical stable Markov operators on ordered Banach spaces. In terms of the er-godicity coefficient, we prove the equivalence of uniform and weak mean ergodicities ofMarkov operators. This result allowed us to establish a category theorem for uniformlymean ergodic Markov operators. Furthermore, following some ideas of [11, 13] and us-ing properties of δ ( T ), we develop the perturbation theory for uniformly asymptoticalstable Markov chains in the abstract scheme. Our results open new perspectives inthe perturbation theory for quantum Markov processes in more general von Neumannalgebras setting, which have significant applications in quantum theory [18].2. Preliminaries
In this section we recall some necessary definitions and fact about ordered Banachspaces.Let X be an ordered vector space with a cone X + = { x ∈ X : x ≥ } . A subset K is called a base for X , if one has K = { x ∈ X + : f ( x ) = 1 } for some strictly positive(i.e. f ( x ) > x >
0) linear functional f on X . An ordered vector space X withgenerating cone X + (i.e. X = X + − X + ) and a fixed base K , defined by a functional f , is called an ordered vector space with a base [2]. In what follows, we denote it as( X, X + , K , f ). Let U be the convex hull of the set K ∪ ( −K ), and let k x k K = inf { λ ∈ R + : x ∈ λU } . Then one can see that k · k K is a seminorm on X . Moreover, one has K = { x ∈ X + : k x k K = 1 } , f ( x ) = k x k K for x ∈ X + . If the set U is linearly bounded (i.e. for anyline ℓ the intersection ℓ ∩ U is a bounded set), then k · k K is a norm, and in this case( X, X + , K , f ) is called an ordered normed space with a base . When X is complete withrespect to the norm k · k K and the cone X + is closed, then ( X, X + , K , f ) is called anordered Banach space with a base (OBSB) . In the sequel, for the sake of simplicityinstead of k · k K we will use usual notation k · k .Let us provide some examples of OBSB.1. Let M be a von Neumann algebra. Let M h, ∗ be the Hermitian part of thepredual space M ∗ of M . As a base K we define the set of normal states of M . Then ( M h, ∗ , M ∗ , + , K , I) is a OBSB, where M ∗ , + is the set of all positivefunctionals taken from M ∗ , and I is the unit in M .2. Let X = ℓ p , 1 < p < ∞ . Define X + = (cid:26) x = ( x , x , . . . , x n , . . . ) ∈ ℓ p : x ≥ (cid:18) ∞ X i =1 | x i | p (cid:19) /p (cid:27) and f ( x ) = x . Then f is a strictly positive linear functional. In this case, wedefine K = { x ∈ X + : f ( x ) = 1 } . Then one can see that ( X, X + , K , f ) is aOBSB. Note that the norm k · k K is equivalent to the usual ℓ p -norm. NIFORM ERGODICITY AND PERTURBATION BOUNDS 3
Let (
X, X + , K , f ) be an OBSB. It is well-known (see [2, Proposition II.1.14]) thatevery element x of OBSB admits a decomposition x = y − z , where y, z ≥ k x k = k y k + k z k . From this decomposition, we obtain the following fact. Lemma 2.1. [15]
For every x, y ∈ X such that x − y ∈ N there exist u, v ∈ K with x − y = k x − y k u − v ) . Let (
X, X + , K , f ) be an OBSB. A linear operator T : X → X is called positive, if T x ≥ x ≥
0. A positive linear operator T : X → X is called Markov , if T ( K ) ⊂ K . It is clear that k T k = 1, and its adjoint mapping T ∗ : X ∗ → X ∗ acts inordered Banach space X ∗ with unit f , and moreover, one has T ∗ f = f . Note that incase of X = R n , X + = R n + and K = { ( x i ) ∈ R n : x i ≥ , P ni =1 x i = 1 } , then for anyMarkov operator T acting on R n , the conjugate operator T ∗ can be identified with ausual stochastic matrix. Now for each y ∈ X we define a linear operator T y : X → X by T y ( x ) = f ( x ) y . For a given operator T we denote A n ( T ) = 1 n n − X k =0 T k , n ∈ N . Definition 2.2.
A Markov operator T : X → X is called(i) uniformly asymptotically stable if there exist an element y ∈ K such thatlim n →∞ k T n − T y k = 0;(ii) uniformly mean ergodic if there exist an element y ∈ K such thatlim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) A n ( T ) − T y (cid:13)(cid:13)(cid:13)(cid:13) = 0;(iii) weakly ergodic if one haslim n →∞ sup x,y ∈K k T n x − T n y k = 0;(iv) weakly mean ergodic if one haslim n →∞ sup x,y ∈K k A n ( T ) x − A n ( T ) y k = 0 . Remark . We notice that uniform asymptotical stability implies uniform mean er-godicity. Moreover, if T is uniform mean ergodic, then y , corresponding to T y , is afixed point of T . Indeed, taking limit in the equality (cid:18) n (cid:19) A n +1 ( T ) − n I = T A n ( T )we find T T y = T y , which yields T y = y . We stress that every uniformly meanergodic Markov operator has a unique fixed point.Let ( X, X + , K , f ) be an OBSB and T : X → X be a linear bounded operator. Letting(2.1) N = { x ∈ X : f ( x ) = 0 } , we define(2.2) δ ( T ) = sup x ∈ N, x =0 k T x kk x k . The quantity δ ( T ) is called the Dobrushin’s ergodicity coefficient of T (see [15]). NAZIFE ERKURS¸UN ¨OZCAN, FARRUKH MUKHAMEDOV
Remark . We note that if X ∗ is a commutative algebra, the notion of the Do-brushin’s ergodicity coefficient was studied in [4],[5] (see [10, 20] for review). In anon-commutative setting, i.e. when X ∗ is a von Neumann algebra, such a notion wasintroduced in [14]. We should stress that such a coefficient has been independentlydefined in [8]. Furthermore, for particular cases, i.e. in a non-commutative setting, thecoefficient explicitly has been calculated for quantum channels (i.e. completely positivemaps).The next result establishes several properties of the Dobrushin’s ergodicity coeffi-cient. Theorem 2.5. [15]
Let ( X, X + , K , f ) be an OBSB and T, S : X → X be Markovoperators. The following assertions hold: (i) 0 ≤ δ ( T ) ≤ ; (ii) | δ ( T ) − δ ( S ) | ≤ δ ( T − S ) ≤ k T − S k ; (iii) δ ( T S ) ≤ δ ( T ) δ ( S ) ; (iv) if H : X → X is a linear bounded operator such that H ∗ ( f ) = 0 , then k T H k ≤ δ ( T ) k H k ; (v) one has δ ( T ) = 12 sup u,v ∈K k T u − T v k ;(vi) if δ ( T ) = 0 , then there exists y ∈ X + such that T = T y .Remark . Note that taking into account Theorem 2.5(v) we obtain that the weakergodicity (resp. weak mean ergodicity) is equivalent to the condition δ ( T n ) → δ ( A n ( T )) →
0) as n → ∞ .The following theorem gives us the conditions that are equivalent to the uniformasymptotical stability. Theorem 2.7. [15]
Let ( X, X + , K , f ) be an OBSB and T : X → X be a Markovoperator. The following assertions are equivalent: (i) T is weakly ergodic; (ii) there exists ρ ∈ [0 , and n ∈ N such that δ ( T n ) ≤ ρ ; (iii) T is uniformly asymptotically stable. Moreover, there are positive constants C, α, n ∈ N and x ∈ K such that k T n − T x k ≤ Ce − αn , ∀ n ≥ n . Uniform mean ergodicity
In this section, we are going to establish an analogous of Theorem 2.7 for uniformlymean ergodic Markov operators.Let (
X, X + , K , f ) be an OBSB. By U we denote the set of all Markov operators from X to X which have an eigenvalue 1 and the corresponding eigenvector f belongs to K . Theorem 3.1.
Let ( X, X + , K , f ) be an OBSB and T ∈ U . Then the following state-ments are equivalent: (i) T is weakly mean ergodic; (ii) There exist ρ ∈ [0 , and n ∈ N such that δ ( A n ( T )) ≤ ρ ; (iii) T is uniformly mean ergodic. NIFORM ERGODICITY AND PERTURBATION BOUNDS 5
Proof.
The implications ( i ) ⇒ ( ii ) and ( iii ) ⇒ ( i ) are obvious. It is enough to provethe implication ( ii ) ⇒ ( iii ).Let us assume that there exist n ∈ N and ρ ∈ [0 ,
1) such that δ ( A n ( T )) ≤ ρ .Since T is Markov operator on X we have k A n ( T )( I − T ) k ≤ n (1 + k T k )and for each k ∈ N (cid:13)(cid:13)(cid:13) A n ( T )( I − T k ) (cid:13)(cid:13)(cid:13) ≤ n (1 + k T k + · · · + (cid:13)(cid:13)(cid:13) T k − (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13) T n − (cid:13)(cid:13) + · · · + (cid:13)(cid:13)(cid:13) T k − n +1 (cid:13)(cid:13)(cid:13) ) . Hence, both norms converges to zero as n → ∞ . Therefore, for each m ∈ N one getslim n →∞ k A n ( T )( I − A m ( T )) k = lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A n ( T ) (cid:18) m m − X k =0 ( I − T k ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m m − X k =0 A n ( T )( I − T k )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0which implies(3.1) lim n →∞ δ ( A n ( T )( I − A m ( T )) = 0 . From (ii) Theorem 2.5 one finds | δ ( A n ( T ) A n ( T )) − δ ( A n ( T )) | ≤ δ ( A n ( T )( I − A n ( T )) . From this inequality with (iii) Theorem 2.5 we infer that δ ( A n ( T )( I − A n ( T )) ≥ δ ( A n ( T )) − δ ( A n ( T ) A n ( T )) ≥ δ ( A n ( T )) − δ ( A n ( T )) δ ( A n ( T )) ≥ (1 − ρ ) δ ( A n ( T ))(3.2)So, from (3.1) and (3.2) we obtain lim n →∞ δ ( A n ( T )) = 0, i.e.lim n →∞ sup x,y ∈K k A n ( T ) x − A n ( T ) y k = 0 . (3.3)Due to T ∈ U one can find a fixed point y ∈ K of T , which from (3.3) yieldslim n →∞ sup x ∈K k A n ( T ) x − y k ≤ lim n →∞ sup x,y ∈K k A n ( T ) x − A n ( T ) y k = lim n →∞ δ ( A n ( T )) = 0 . which means the uniform mean ergodicity of T . This completes the proof. (cid:3) By U ume we denote the set of all uniformly mean ergodic Markov operators belongingto U . Theorem 3.2.
Let ( X, X + , K , f ) be an OBSB. Then the set U ume is a norm dense andopen subset of U .Proof. Take an arbitrary T ∈ U with a fixed point φ ∈ K . Let 0 < ε < T ( ε ) = (cid:18) − ε (cid:19) T + ε T φ . NAZIFE ERKURS¸UN ¨OZCAN, FARRUKH MUKHAMEDOV
It is clear that T ( ε ) ∈ U , since T ( ε ) φ = φ , and k T − T ( ε ) k < ε . Now we show that T ( ε ) ∈ U ume . It is enough to establish that T ( ε ) is uniform asymptotically stable (seeRemark 2.3). Indeed, by Lemma 2.1, if x − y ∈ N , we get k T ( ε ) ( x − y ) k = k x − y k k T ( ε ) ( u − v ) k = k x − y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − ε (cid:19) T ( u − v ) + ε T φ ( u − v ) (cid:13)(cid:13)(cid:13)(cid:13) = k x − y k (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − ε (cid:19) T ( u − v ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:18) − ε (cid:19) k x − y k which implies δ ( T ( ε ) ) ≤ − ε . Here u, v ∈ K . Hence, due to Theorem 2.7 we infer that T ( ε ) is uniform asymptotically stable.Now let us show that U ume is a norm open set. First, for each n ∈ N , we define U ume,n = (cid:26) T ∈ U : δ ( A n ( T )) < (cid:27) . Then one can see that U ume = [ n ∈ N U ume,n . Therefore, to establish the assertion, it is enough prove that U ume,n is a norm open set.Take any T ∈ U ume,n , and put α := δ ( A n ( T )) <
1. Choose 0 < β < α + β <
1. Let us show that (cid:26) H ∈ U : k H − T k < βn + 1 (cid:27) ⊂ U ume,n . We note that for each k ∈ N one has k H k − T k k ≤ k H k − ( H − T ) k + k ( H k − − T k − ) T k≤ k H − T k + k H k − − T k − k· · ·≤ k k H − T k . (3.4)From (ii) of Theorem 2.5 with (3.4) we find | δ ( A n ( H )) − δ ( A n ( T )) | ≤ k A n ( H ) − A n ( T ) k≤ n n − X k =0 k H k − T k k≤ n n X k =1 k k H − T k = n + 12 k H − T k < β NIFORM ERGODICITY AND PERTURBATION BOUNDS 7
Hence, the last inequality yields that δ ( A n ( H )) < δ ( A n ( T )) + β <
1. This due toTheorem 3.1 implies H ∈ U ume,n . This completes the proof. (cid:3) Corollary 3.3.
Let T ∈ U be a uniformly mean ergodic Markov operator. Then there isa neighborhood of T in U such that every Markov operator taken from that neighborhoodhas a unique fixed point.Remark . We point out that the question on the geometric structure of the setof uniformly ergodic operators was initiated in [9]. The proved theorem gives someinformation about the set of uniformly mean ergodic operators.4.
Perturbation Bounds and Uniform asymptotic stability of Markovoperators
In this section, we prove perturbation bounds in terms of C and e α under the con-dition k T n − T x k ≤ Ce − αn . Moreover, we also give several bounds in terms of theDobrushin’s ergodicity coefficient. Theorem 4.1.
Let ( X, X + , K , f ) be an ordered Banach space with a base, and S , T beMarkov operators on X . If T is uniformly asymptotically stable, then one has k T n x − S n z k ≤ (4.1) ( k x − z k + n k T − S k , ∀ n ≤ ˜ n,Ce − αn k x − z k + (˜ n + C e − α ˜ n − e − αn − e − α ) k T − S k , ∀ n > ˜ n where ˜ n := log (cid:20) log(1 /C ) e − α (cid:21) , C ∈ R + , α ∈ R + , x, z ∈ K .Proof. For each n ∈ N , by induction we have(4.2) S n = T n + n − X i =0 T n − i − ◦ ( S − T ) ◦ S i . Let x, z ∈ K it then follows from (4.2) that T n x − S n z = T n x − T n z − n − X i =0 T n − i − ◦ ( S − T ) ◦ S i ( z )= T n ( x − z ) − n − X i =0 T n − i − ◦ ( S − T )( z i ) , where z i = S i z . Hence, k T n x − S n z k ≤ k T n ( x − z ) k + n − X i =0 (cid:13)(cid:13) T n − i − ◦ ( S − T )( z i ) (cid:13)(cid:13) . Since T and S are Markov operator and due to (iv) of Theorem 2.5 one finds (cid:13)(cid:13) T n − i − ◦ ( S − T )( z i ) (cid:13)(cid:13) ≤ δ ( T n − i − ) k S − T k and k T n ( x − z ) k ≤ δ ( T n ) k x − z k . NAZIFE ERKURS¸UN ¨OZCAN, FARRUKH MUKHAMEDOV
Hence, we obtain k T n x − S n z k ≤ δ ( T n ) k x − z k + n − X i =0 δ ( T n − i − ) k S − T k = δ ( T n ) k x − z k + k S − T k n − X i =0 δ ( T i ) . (4.3)From (v) Theorem 2.5 one gets δ ( T i ) = 12 sup u,v ∈K (cid:13)(cid:13) T i u − T i v (cid:13)(cid:13) ≤ sup u ∈K (cid:13)(cid:13) T i u − T x u (cid:13)(cid:13) Therefore, due to Theorem 2.7 we have δ ( T n ) ≤ ( , ∀ n ≤ ˜ n,Ce − αn , ∀ n > ˜ n (4.4)where ˜ n = (cid:20) log(1 /C )log e − α (cid:21) = [log C α ].So, from (4.4) we obtain n − X i =0 δ ( T i ) = ˜ n − X i =0 δ ( T i ) + n − X i =˜ n δ ( T i ) ≤ ˜ n + n − X i =˜ n Ce − αi = ˜ n + Ce − a ˜ n − e − α ( n − ˜ n ) − e − α , ∀ n > ˜ n. (4.5)Hence, the last inequality with (4.4) and (4.5) yields the required assertion. (cid:3) Corollary 4.2.
Let ( X, X + , K , f ) be an ordered Banach space with base and S , T beMarkov operators on X . If T is uniformly asymptotically stable to T x , then for every x, y ∈ K one has (4.6) sup n ∈ N k T n x − S n z k ≤ k x − z k + (cid:18) ˜ n + C e − α ˜ n − e − α (cid:19) k T − S k . In addition, if S is uniformly asymptotically stable to S z , then (4.7) k T x − S z k ≤ (cid:18) ˜ n + C e − α ˜ n − e − α (cid:19) k T − S k . Proof.
The inequality (4.6) is a direct consequence of (4.1). Now if we consider (4.3)then one has k T n − S n k = sup x,z ∈K k T n x − S n z k≤ δ ( T n ) sup x,z ∈K k x − z k + n − X i =0 δ ( T n − i − ) k T − S k and taking the limit as n → ∞ one finds k T x − S z k ≤ k T − S k ∞ X i =0 δ ( T i ) . NIFORM ERGODICITY AND PERTURBATION BOUNDS 9
From (4.5) it follows that k T x − S z k ≤ k T − S k (cid:18) ˜ n + C e − α ˜ n − e − α (cid:19) . This completes the proof. (cid:3)
The inequality (4.3) allows us to obtain perturbation bounds in terms of the Do-brushin’s coefficient of T . Namely, we have the following result. Theorem 4.3.
Let ( X, X + , K , f ) be an ordered Banach space with base and S , T beMarkov operators on X . If there exists a positive integer m such that δ ( T m ) < (i.e. T is uniformly asymptotically stable), then for every x, z ∈ K one has (4.8) sup k ∈ N (cid:13)(cid:13)(cid:13) T km x − S km z (cid:13)(cid:13)(cid:13) ≤ δ ( T m ) k x − z k + k T m − S m k − δ ( T m ) and k T n x − S n z k ≤ k x − z k + max
By the inequality (4.3) for every x, z ∈ K we obtainsup n ∈ N k T n x − S n z k ≤ δ ( T n ) k x − z k + k T − S k − δ ( T )and if S is also uniformly asymptotically stable to S z then one gets k T x − S z k ≤ k T − S k − δ ( T ) . If we consider T m instead of T , then one finds the inequalities (4.8) and (4.10).Now for every k ∈ N and every integer i such that 1 ≤ i ≤ m −
1, we have T mk + i x − S mk + i z = ( T mk − S mk ) T i x + S mk ( T i x − S i z ) . Therefore,(4.11) (cid:13)(cid:13)(cid:13) T mk + i x − S mk + i z (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) T mk − S mk (cid:13)(cid:13)(cid:13) + δ k ( S m ) (cid:13)(cid:13) T i x − S i z (cid:13)(cid:13) . Due to T n x − S n z = S n ( x − z ) + ( T n − S n ) x , we get(4.12) k T n x − S n z k ≤ k x − z k + k T n − S n k where n < m .If n ≥ m combining of (4.8) and (4.11)-(4.12) one finds (4.9), which completes theproof. (cid:3) The following theorem gives an alternative method of obtaining perturbation boundsin terms of δ ( T m ). Theorem 4.4.
Let δ ( T m ) < hold for some m ∈ N . Then for every x, z ∈ K one has k T n x − S n z k ≤ δ ( T m ) ⌊ n/m ⌋ ( k x − z k + max
Corollary 4.5.
Let the condition of Theorem 4.4 be satisfied. Then for every x, z ∈ K we have (4.14) sup n ∈ N k T n x − S n z k ≤ sup n ∈ N δ ( T m ) ⌊ n/m ⌋ + m k T − S k − δ ( T m ) . Proof.
Since T i − S i = T ( T i − − S i − ) + ( T − S ) S i − by induction we obtain(4.15) max
Proof.
First we prove that the operator ( I − S m ) − is bounded on the set N (see (2.1)).Indeed, take any x ∈ N , then we have(4.17) k S m x k ≤ k S m − T m x k + k T m x k ≤ ρ k x k . where ρ = k S m − T m k + δ ( T m ) <
1. Hence by (4.17) one gets k S mn x k ≤ ρ n k x k for all n ∈ N . Therefore, the series P n S mn x converges. Using the standard technique, onecan see that ( I − S m ) − x = X n S mn x and moreover, k ( I − S m ) − x k ≤ k x k − ρ , for all x ∈ N . This means that ( I − S m ) − isbounded on N .It is clear that the equation S m z = z with z ∈ K equivalent to ( I − S m )( z − x ) = − ( I − S m ) x . Due to ( I − S m ) x ∈ N we conclude the last equation has a unique solution z = x − ( I − S m ) − (( I − S m ) x ) . From the identity z − x = T m ( z − x ) + ( S m − T m )( z − x ) + ( S m − T m ) x and keeping in mind z − x ∈ N one finds k z − x k ≤ (cid:0) δ ( T m ) + k S m − T m k (cid:1) k z − x k + k S m − T m k which implies (4.16).From S m ( Sz ) = S ( S m z ) = Sz , and the uniqueness of z for S m we infer that Sz = z . Now assume that S has another fixed point ˜ z ∈ K . Then S m ˜ z = ˜ z which yields ˜ z = z . Moreover, due to (4.17) one concludes that δ ( S m ) <
1, whichby Theorem 2.7 yields that S is uniformly asymptotically stable. This completes theproof. (cid:3) Remark . The obtained results can be applied in several directions.(i) We note that all obtained results extend main results of [11, 13, 18] to generalBanach spaces. Hence, they allow to apply the obtained estimates to Markovchains over various spaces.(ii) Considering the classical L p -spaces, one may get the perturbation bounds foruniformly asymptotically stable Markov chains defined on these L p -spaces. Onthe other hand, one may directly apply the results to Markov chains defined onmore complicated functional spaces. Moreover, by varying the Banach space onecan obtain several interesting results in the theory of measure-valued Markovprocesses.(iii) All obtained results are even new, if one takes X as pre-duals of either von Neu-mann algebra or J BW -algebra. Moreover, if we take X as a dual of C ∗ -algebras,then one gets interesting perturbation bounds for strong mixing C ∗ -dynamicalsystems. If we consider non-commutative L p -spaces, then the perturbationbounds open new perspectives in the quantum information theory (see [18]). Acknowledgments
The first author (N.E.) thanks Hacettepe University Scientific Research ProjectsCoordination Unit support this project under the Project Number: 014 D12 601 005 -832. She is also grateful to International Islamic University Malaysia for kind hospitalityof her research stay and this work is started there. The second named author (F.M.) thanks also Hacettepe University (Turkey) for kind hospitality during 5-9 September2015, where a part of this work is carried out.
References
1. S.Albeverio, R.Høegh-Krohn, Frobenius theory for positive maps of von Neumann algebras,
Comm.Math. Phys. (1978), 83–94.2. E.M. Alfsen, Compact convex sets and booundary integrals , Springer-Verlag, Berlin, (1971).3. W. Bartoszek, Asymptotic properties of iterates of stochastic operators on (AL) Banach lattices,
Anal. Polon. Math. (1990), 165-173.4. J. E. Cohen, Y. Iwasa, G. Rautu, M.B. Ruskai, E. Seneta, G. Zbaganu, Relative entropy undermappings by stochastic matrices, Linear Algebra Appl. (1993), 211-235.5. R. L. Dobrushin, Central limit theorem for nonstationary Markov chains. I,II,
Theor. Probab. Appl. (1956),65–80; 329–383.6. E. Yu. Emel’yanov, M.P.H. Wolff, Positive operators on Banach spaces ordered by strongly normalcones, Positivity (2003), 3–22.7. F. Fagnola, R. Rebolledo, On the existance of stationary states for quantum dyanamical semigroups, Jour. Math. Phys. (2001), 1296–1308.8. S. Gaubert, Z. Qu, Dobrushin’s ergodicity coefficient for Markov operators on cones and beyond, Integ. Eqs. Operator Theor. (2014), 127–150.9. P.R. Halmos, Lectures on erodic theory , Chelsea, New York, 1960.10. I.C.F. Ipsen, T.M. Salee, Ergodicity coefficients defined by vector norms,
SIAM J. Matrix Anal.Appl. (2011), 153–200.11. N.V. Kartashov, Inequalities in theorems of ergodicity and stability for Markov chains with commonPhase space, I, Probab. Theor. Appl. (1986), 247–259.12. U. Krengel, Ergodic Theorems , Walter de Gruyter, Berlin-New York, 1985.13. A. Mitrophanov, Sensitivty and convergence of uniform ergodic Markov chains,
J. Appl. Probab. (2005), 1003–1014.14. F. Mukhamedov, Dobrushin ergodicity coefficient and ergodicity of noncommutative Markov chains, J. Math. Anal. Appl. (2013), 364–373.15. F. Mukhamedov, Ergodic properties of nonhomogeneous Markov chains defined on ordered Banachspaces with a base,
Acta. Math. Hungar. (2015), 294–323.16. F. Mukhamedov, Strong and weak ergodicity of nonhomogeneous Markov chains defined on orderedBanach spaces with a base,
Positivity (2016), 135–153.17. C. Niculescu, A. Str¨oh, L. Zsid´o, Noncommutative extensions of classical and multiple recurrencetheorems, J. Operator Theory (2003), 3–52.18. D. Reeb, M. J. Kastoryano, M. M. Wolf, Hilbert’s projective metric in quantum information theory, J. Math. Phys. (2011), 082201.19. T.A. Sarymsakov, N.P. Zimakov, Ergodic principle for Markov semi-groups in ordered normalspaces with basis, Dokl. Akad. Nauk. SSSR (1986), 554–558.20. E. Seneta,
Non-negative matrices and Markov chains , Springer, Berlin, 2006.21. O. Szehr, M.M. Wolf, Perturbation bounds for quantum Markov processes and their fixed points,
J. Math. Phys. (2013), 032203. Department of Mathematics, Faculty of Science, Hacettepe University, Ankara,06800,Turkey.
E-mail address : [email protected] Department of Computational and Theoretical Science, Faculty of Science, Inter-national Islamic University Malaysia, Kuantan, Pahang, 25710, Malaysia.
E-mail address ::