Infinite Dimensional Orthogonal Preserving Quadratic Stochastic Operators
aa r X i v : . [ m a t h . F A ] A p r INFINITE DIMENSIONAL ORTHOGONAL PRESERVING QUADRATICSTOCHASTIC OPERATORS
FARRUKH MUKHAMEDOV AND AHMAD FADILLAH EMBONG
Abstract.
In the present paper, we study infinite dimensional orthogonal preserving qua-dratic stochastic operators (OP QSO). A full description of OP QSOs in terms of their canon-ical form and heredity coefficient’s values is provided. Furthermore, some properties of OPQSOs and their fixed points are studied.
Mathematics Subject Classification : 46L35, 46L55, 46A37.
Key words : quadratic stochastic operators; orthogonal preserving; infinite dimensional. Introduction
The history of quadratic stochastic operators (QSOs) is traced back to Bernstein’s work[4] where such kind of operators appeared from the problems of population genetics (see also[14]). These kind of operators describe time evolution of variety species in biology and arerepresented by so-called Lotka-Volterra(LV) systems [23], but currently in the present, thereare many papers devoted to these operators owing to the fact that they have plentiful appli-cations especially in modelings in many different fields such as biology [11, 19] (populationand disease dynamics), physics [20, 22](non-equilibrium statistical mechanics) , economics, andmathematics [14, 19, 22] (replicator dynamics and games).A quadratic stochastic operator is usually used to present the time evolution of species inbiology, which arises as follows. By considering an evolution of species in biology as given in thesituation where I = { , , . . . , n } is the n type of species (or traits) in a population, the proba-bility distribution of the species in an early state of that population is x (0) = ( x (0)1 , . . . , x (0) n ). Ona side note, we define P ij,k as the probability of an individual in the i th species and j th species tocross-fertilize and produce an individual from k th species (trait). Given x (0) = ( x (0)1 , . . . , x (0) n ),we can find the probability distribution of the first generation, x (1) = ( x (1)1 , . . . , x (1) n ) by using atotal probability, i.e., x (1) k = n X i,j =1 P ij,k x (0) i x (0) j , k ∈ { , . . . , n } . This relation defines an operator which is denoted by V and it is called quadratic stochasticoperator (QSO) . Each QSO maps the simplex S n − = { x ∈ R n | x i ≥ , n P i =1 x i = 1 } intoitself. Moreover, the operator V can be interpreted as an evolutionary operator that describesthe sequence of generations in terms of probability distributions if the values of P ij,k and thedistribution of the current generation are given. The most well-known class in the theory QSOis a Volterra one, namely whose heredity coefficients satisfy P ij,k = 0 if k / ∈ { i, j } . (1.1) The condition (1.1), biologically, means that each individual can inherit only the species of theparents. The dynamics of Volterra QSO was studied in [9, 8]. Nevertheless, not all QSOs are ofVolterra-type, therefore, the understanding of the dynamics of non-Volterra QSO still remainsopen. We refer the reader to [10, 17] as the exposition of the recent achievements and openproblems in the theory of the QSO can be further researched.One of the main problems in the theory of nonlinear operator is to study the limiting behaviorof nonlinear operators. To this day, there are a handful of studies dedicated to the explorationof the dynamics of higher dimensional systems despite the fact that it is a very exquisite andimportant topic. Although, most research has been focused on the simplex S n − , but there aremodels where the probability distribution is given on a countable set, which means that thecorresponding QSO is defined on an infinite-dimensional space.The simplest case of the infinite-dimensional space is the Banach space ℓ of absolutelysummable sequences. It is worth mentioning that some infinite dimensional QSOs were studiedin [13, 15, 16].On the other hand, from [21] with the results of [18] we conclude that a QSO (acting on finitedimensional simplex) is surjective, if and only if, it is orthogonal preserving (OP) QSO. Here bythe orthogonality of distributions we mean their disjointness. We cannot afford to ignore thesurjectivity of a quadratic operator is strongly tied up with nonlinear optimization problems[3]. Furthermore, any orthogonal preserving QSO is a permutation of Volterra QSO in[1, 18].Yet, if we look at the same problem in the infinite dimensional setting, the last statementbecomes incorrect. Also in [1], we have considered a special class of orthogonal preservingoperators for which an analogous result was obtained replicated in the finite dimensional setting.Unfortunately, this type of result is wrong in a general setting. Therefore, in this paper, we go ona voyage of discovery in an attempt to describe the orthogonality preserving infinite dimensionalquadratic stochastic operators in a general case. We notice that every linear stochastic operatorscan be considered as a particular case of QSO. In the later case, there are many papers tha aredevoted to the orthogonal preserving linear operators defined on various Banach spaces (seefor example [2, 5, 6, 7, 12, 24]), once the nonlinearity appears in operators, then all existingmethods (for linear operators) are no longer applicable. The simplest nonlinearity is quadraticwhich for these kinds of we fully describe, OP QSOs in terms of the their heredity coefficients,and provide their canonical forms. Last but not least, we provide ceratin examples of such kindof operators along with the properties of OP QSOs and their fixed points.2. Orthogonal Preserving QSO
Let E be a subset of N . Denote S E = ( x = ( x i ) ∈ E ∈ R E : x i ≥ , X i ∈ E x i = 1 ) . In what follows, by e i we denote the standard basis in S E , i.e. e i = ( δ ik ) k ∈ E ( i ∈ E ), where δ ij is the Kroneker delta.Let V be a mapping defined by V ( x ) k = X i,j ∈ E P ij,k x i x j , k ∈ E (2.1) NFINITENESS OF ORTHOGONAL PRESERVING QSO 3 here, { P ij,k } are hereditary coefficients which satisfy P ij,k ≥ , P ij,k = P ji,k , X k ∈ E P ij,k = 1 , i, j, k ∈ E (2.2)One can see that V maps S E into itself and V is called Quadratic Stochastic Operator (QSO) [15].By support of x = ( x i ) i ∈ E ∈ S E we mean a set Supp ( x ) = { i ∈ E : x i = 0 } . A sequence { A k } of sets is called cover of a set B if ∞ S k =1 A k = B and A i ∩ A j = ∅ for i, j ∈ N ( i = j ).Recall that two vectors x = ( x k ) , y = ( y k ) belonging to S E are called orthogonal (denotedby x ⊥ y ) if supp ( x ) ∩ supp ( y ) = ∅ . If x , y ∈ S E , then one can see that x ⊥ y if and only if x ◦ y = 0 (or x k · y k = 0 for all k ∈ E ). Here, ◦ stands for the standard dot product. Definition 2.1.
A QSO V given by (2.1) is called orthogonal preserving QSO (OP QSO) iffor any x , y ∈ S with x ⊥ y one has V ( x ) ⊥ V ( y ) . Let T be a stochastic matrix given by ( t ij ) i,j ∈ E , where t ij ≥ P j ∈ E t ij = 1 ( i ∈ N ). Thenone can define a linear operator (which is called linear stochastic operator (LSO))( T x ) k = ( x T ) k = X i ∈ E t ik x i , x ∈ S E , k ∈ E. (2.3)Due to stochasticity of T , the operator T maps S E into itself. Note that each LSO can beconsidered as a particular case of q.s.o. Indeed, let us define(2.4) P ( T ) ij,k = t ik + t jk . Then one can see that { P ( T ) ij,k } satisfies (2.2), and for the corresponding q.s.o. V T we have( V T ( x )) k = X i,j ∈ E P ( T ) ij,k x i x j = X i,j ∈ E (cid:18) t ik + t jk (cid:19) x i x j = X i ∈ E t ik x i = ( T x ) k , ∀ k ∈ N , i.e. V ( x ) = T x for all x ∈ S E . This implies that all results holding for QSO are valid for LSO. Remark 2.2.
Let T be a LSO, then for any x = ( x k ) ∈ S E , T can be written as follows T ( x ) = X k ∈ E x k T ( e k ) . Therefore, a LSO T defined on S E is orthogonal preserving if and only if T ( e i ) ⊥ T ( e j ) forall i = j . Indeed, it is enough to show that the last statement implies OP of T . Let us take x , y ∈ S E such that x ⊥ y (i.e. x ◦ y = 0 ). Then from T ( x ) = X ℓ ∈ E x ℓ T ( e ℓ ) , T ( y ) = X m ∈ E y m T ( e m ) FARRUKH MUKHAMEDOV AND AHMAD FADILLAH EMBONG with T ( e ℓ ) ◦ T ( e m ) = δ ℓm we find T ( x ) ◦ T ( y ) = (cid:18) X ℓ ∈ E x ℓ T ( e ℓ ) (cid:19) ◦ (cid:18) X m ∈ E y m T ( e m ) (cid:19) = X ℓ,m ∈ E x ℓ y m T ( e ℓ ) ◦ T ( e m )= X ℓ ∈ E x ℓ y ℓ = 0 Now using (2.3) we conclude that T is OP if and only if for the stochastic matrix ( t ij ) one has t i ⊥ t j for all i, j ∈ N with i = j . Here t k = ( t ki ) i ∈ E , k ∈ E for all i = j . When we consider the QSO, then similar kind of result is not valid, but we use some ideasfrom the mentioned remark.
Remark 2.3.
We first note that if V is an OP QSO, then the system { V ( e k ) } is also orthogonal.Therefore, to describe OP QSO it is enough for us just to fix this (i.e. { V ( e k ) } ) system. Indeed,let us denote by V the set of all OP QSO V such that V ( e k ) = F k for some orthogonal system F k in S . Now, let us assume that an OP QSO e V such that e V (˜ F k ) = F ′ k , where { ˜ F k } and { F ′ k } are orthogonal systems in S . On the other hand, if one considers { e V ( e k ) } then the system isalso has to be orthogonal in S i.e., e V ( e k ) = I k , where { I k } is an orthogonal system in S . Hence e V is an element of V . Recall [16] that a QSO V : S E → S E is called Volterra if one has P ij,k = 0 if k / ∈ { i, j } , i, j, k ∈ E. (2.5) Remark 2.4. In [16] it was given an alternative definition Volterra operator in terms of ex-tremal elements of S E . One can check [17] that a QSO V is Volterra if and only if one has( V ( x )) k = x k X i ∈ E a ki x i ! , k ∈ E, where a ki = 2 P ik,k − i, k ∈ E . One can see that a ki = − a ik . This representation leads us tothe following definitions. Definition 2.5.
A QSO V : S E → S E is called π -Volterra if there is a permutation π of E such that V has the following form V ( x ) k = x π ( k ) X i ∈ E a π ( k ) i x i ! where a π ( k ) i = 2 P iπ ( k ) ,k − , a π ( k ) i = − a iπ ( k ) for any i, k ∈ E . In [1, 18] it has been proved the following result.
Theorem 2.6 ([1, 18]) . Let E = { , , . . . , n } and V be a QSO on S E . Then the followingstatements are equivalent: NFINITENESS OF ORTHOGONAL PRESERVING QSO 5 (i) V is orthogonal preserving; (ii) V is π -Volterra QSO. In what follows, for the sake of convenience we denote S instead of S N . Remark 2.7.
We notice that the vertices of the finite simplex S E ( E = { , , . . . , n } ) aredescribed by the elements e k = ( δ ik ) i ∈ E . Therefore, any OP QSO on S n − is a permutatedVolterra QSO (see Theorem 2.6). However, if we consider S , then one can see that there aremany orthogonal systems in S , which differ from the system { e k } . For example F (1 / = (cid:18) , , , . . . (cid:19) , F (1 / = (cid:18) , , , , , . . . (cid:19) ,. . . , F (1 / k = , , . . . , |{z} k − , |{z} k , , . . . , . . . (2.6) Another crucial moment is that for a given orthogonal system { F k } in S , the set ∞ [ k ( supp ( F k )) may not equal to N . For example, we have ∪ ∞ k =2 ( supp ( e k )) = N \ { } . All these make thedescription of OP QSOs is more challenging than the finite dimensional setting. In [1] a special class of infinite dimensional OP QSOs have been studied for which an analo-gous of Theorem 2.6 holds.
Theorem 2.8 ([1]) . Let V be a QSO on S such that V ( e i ) = e π ( i ) for some permutation π : N → N . Then V is an OP QSO if and only if V is π -Volterra QSO. Recall that an orthogonal basis { F k } ∞ k =1 in S is called total if for any x ∈ S one finds { λ i } ∞ i =1 , λ ≥ , P ∞ i =1 λ i = 1 such that x = ∞ X i =1 λ i F i Theorem 2.9.
Let { F k } ∞ k =1 be an orthogonal basis in S . The following conditions are equivalent (i) { F i } ∞ i =1 is total; (ii) For every k ∈ N one has | supp ( F k ) | = 1 and ∪ ∞ k =1 supp ( F k ) = N .Proof. (i) ⇒ (ii). Assume contrary i.e., there exists some k ∈ N such that | supp ( F k ) | ≥ m ∈ supp ( F k ). If one considers e m ∈ S , then due to the totality of { F k } we have e m = X i λ i F i This means that λ i = 0 for i = k , so e m = F k , which contradicts to | supp ( F k ) | ≥
2. Now,if ∪ ∞ k =1 supp ( F k ) ⊂ N , then for ℓ ∈ N \ ∪ ∞ k =1 supp ( F k ), the vector e ℓ can not be represent as aconvex combination of { F k } . Hence, we infer the statement (ii).(ii) ⇒ (i). If (ii) holds, then the system { F k } ∞ k =1 is a permutation of the standard basis { e k } ∞ k =1 , which is clearly total. (cid:3) FARRUKH MUKHAMEDOV AND AHMAD FADILLAH EMBONG
From the last theorem and Theorem 2.8 we conclude the following result.
Corollary 2.10.
Let { F k } ∞ k =1 be an orthogonal system in S and V is an OP QSO on S suchthat V ( e k ) = F k , for all k ∈ N . Then the following statements are equivalent (i) V is an π − Volterra QSO; (ii) F k is total. Description of OP QSOs
In this section, we are going to describe infinite dimensional OP QSOs.Let V be a QSO on S whose heredity coefficients are { P ij,k } . Let us introduce the followingvectors P ij = ( P ij, , · · · , P ij,n , · · · ) for any i, j ∈ N One can see that for every i, j ∈ N the vector P ij belongs to S . Next result describes OP QSOsin terms of the vectors { P ij } . Theorem 3.1.
Let V be a QSO. Then the following conditions are equivalent: (i) V is an OP QSO; (ii) For any
A, B ⊂ N with A ∩ B = ∅ one has P ij ⊥ P uv for all i, j ∈ A and u, v ∈ B .Proof. (i) ⇒ (ii). Take any A, B ⊂ N with A ∩ B = ∅ . Then chose two elements x , y ∈ S suchthat supp ( x ) = A and supp ( y ) = B . From the condition A ∩ B = ∅ one concludes that x ⊥ y .From the definition of QSO, we have V ( x ) = X i,j ∈ supp ( x ) P ij,k x i x j ∞ k =1 , V ( y ) = P u,v ∈ supp ( y ) P uv,k y u y v ! ∞ k =1 . (3.1)Due to the orthogonal preserving property of V one has V ( x ) ◦ V ( y ) = 0, therefore one gets V ( x ) ◦ V ( y ) = ∞ X k =1 X i,j ∈ supp ( x ) P ij,k x i x j X u,v ∈ supp ( y ) P uv,k y u y v = X i,j ∈ supp ( x ) X u,v ∈ supp ( y ) ∞ X k =1 P ij,k P uv,k ! x i x j y u y v = 0(3.2)According to i, j ∈ supp ( x ) and u, v ∈ supp ( y ) (i.e., x i > , y u > i ∈ supp ( x ) and u ∈ supp ( y )) from the last equalities, we conclude that ∞ X k =1 P ij,k P uv,k = 0which means P ij ◦ P uv = 0 for all i, j ∈ A and u, v ∈ B .Now let us prove (ii) ⇒ (i). Now, take x , y ∈ S such that x ⊥ y , then from (3.2) one finds V ( x ) ◦ V ( x ) = X i,j ∈ supp ( x ) X u,v ∈ supp ( y ) ( P ij ◦ P uv ) x i x j y u y v (3.3) NFINITENESS OF ORTHOGONAL PRESERVING QSO 7
Due the fact supp ( x ) ∩ supp ( y ) = ∅ and the assumption (ii) we immediately obtain V ( x ) ◦ V ( x ) =0, i.e. V ( x ) ⊥ V ( x ). This completes the proof. (cid:3) From this theorem we immediately get the following corollary.
Corollary 3.2.
Let V be an OP QSO, then for any i = j ( i, j ∈ N ) one has P ii ⊥ P jj . Remark 3.3.
If a QSO V is given by a stochastic matrix (see (2.4) ) then from Corollary 3.2we infer that V is OP if and only if t i ⊥ t j for all i, j ∈ N ( i = j ). This recovers the result ofRemark 2.2. One can infer that from Theorem 3.1 it is difficult to write representation of OP QSO.Therefore, for a given OP QSO V we denote F k = V ( e k ), k ∈ N . The system F = { F k } isorthogonal. In what follows, we denote F k = ( f k,i ) i ∈ N . One can see that f k,i = 0 if i / ∈ supp ( F k ).Henceforth, | A | is referred to the cardinality of a set A and denote C F = N \ (cid:18) [ k ∈ N supp ( F k ) (cid:19) Theorem 3.4.
Let F = { F k } be an orthogonal system and V be a QSO on S such that V ( e k ) = F k , k ∈ N . Then, V ( x ) is an OP QSO if and only if it has the following form: forany x ∈ S (a) for any m ∈ supp ( F k ) ( V ( x )) m = x k f k,m + ∞ X i =1 a ( m ) ik x i ! where a ( m ) ik = 2 P ik,m − f k,m and set a ( m ) kk = 0 . (b) for any c ∈ C F , V ( x ) c takes one of the following form (I) if there is no P ij,c > for every i, j ∈ N , then V ( x ) c = 0 or (II) if there exists at least one P i c j c ,c > , then V ( x ) c has one of the following form: (i) if there is no P ij,c > for j ∈ { i c , j c } where i ∈ N \{ j } , then V ( x ) c = 2 P i c j c ,c x i c x j c (ii) if there exists P i c j,c > for either j = i c or j = j c (here let j = i c ), then V ( x ) c has one of the following form: (1) if P i c j c ,c > then V ( x ) c = 2 (cid:0) P i c j c ,c x i c x j c + P i c j c ,c x i c x j c + P i c i c ,c x i c x i c (cid:1) (2) if P i c j c ,c = 0 then V ( x ) c = 2 x i c P i c j c ,c x j c + ∞ X i =1 i = i c ,j c P ii c ,c x i Proof.
Let us start with ”if” part, i.e. we assume that V is an OP QSOs. From the assumption V ( e k ) = F k and the definition of QSO we have V ( e k ) = ( P kk, , . . . , P kk,m , . . . ) = F k FARRUKH MUKHAMEDOV AND AHMAD FADILLAH EMBONG
This implies that P kk,m = (cid:26) m / ∈ supp ( F k ) ,f k,m if m ∈ supp ( F k ) , (3.4)By choosing x k = ( x , . . . , x k − , , x k +1 , . . . ) such that x i > , for i ∈ N \{ k } (3.5)and e k one has x k ⊥ e k . Due to the assumption, we infer that V ( x k ) ⊥ V ( e k ). It is clear that V ( x k ) = ∞ X i,j = k P ij, x i x j , . . . , ∞ X i,j = k P ij,m x i x j , . . . ! . Thus, from the fact V ( x k ) ◦ V ( e k ) = 0 and (3.5), we immediately find ∞ X ij = k P ij,m x i x j = 0 ⇒ P ij,m = 0 for any i, j = k and m ∈ supp ( F k ) . Hence, for any m ∈ supp ( F π ( k ) ) and for any x ∈ SV ( x ) m = ∞ X i,j =1 P ij,m x i x j = P kk,m x k + ∞ X i = k P ik,m x i x k + ∞ X j = k P kj,m x j x k Keeping in mind P ik,m = P ki,m , x k = 1 − ∞ P i = k x i and (3.4), ( V ( x )) m reduces to V ( x ) m = x k f k,m + ∞ X i = k (2 P ik,m − f k,m ) x i ! = x k f k,m + ∞ X i =1 a ( m ) ik x i ! which shows (a).Next, let us consider c ∈ C F . Then( V ( x )) c = ∞ X i,j =1 P ij,c x i x j = ∞ X i =1 P ii,c x i + ∞ X i =1 P i ,c x x i + · · · + ∞ X i = n P in,c x n x i + . . . = ∞ X i =1 P ii,c x i + 2 ∞ X i =2 P i ,c x x i + · · · + 2 ∞ X i = n P in,c x n x i + . . . = ∞ X i =1 P ii,c x i + 2 ∞ X j =1 ∞ X i = j +1 P ij,c x i x j Taking into account (3.4), one gets P kk,c = 0 for any k ∈ N and c ∈ C F . Therefore V ( x ) c = 2 ∞ X j =1 ∞ X i = j +1 P ij,c x i x j (3.6) NFINITENESS OF ORTHOGONAL PRESERVING QSO 9
First, we assume that there exist i c , j c ∈ N such that P i c j c ,c > V ( x ) c = 0). Next, let us choose two vectors from the simplex S as follows x ( i c ,j c ) = , . . . , , |{z} i thc , , . . . , , |{z} j thc , y [ i c ,j c ] = ( y , . . . , y i c − , , y i c +1 , . . . , y j c − , , y j c +1 , . . . )where y i > i ∈ N \{ i c , j c } . Clearly x ( i c ,j c ) is orthogonal to y [ i c ,j c ] , hence by assumptionon V V (cid:0) x ( i c ,j c ) (cid:1) ⊥ V (cid:0) y [ i c ,j c ] (cid:1) (3.7)From the part (a), one gets V (cid:0) x ( i c ,j c ) (cid:1) k · V (cid:0) y [ i c ,j c ] (cid:1) k = 0 ∀ k ∈ [ i ∈ N supp ( F i )Using (3.6), one has V (cid:0) x ( i c ,j c ) (cid:1) c = 12 P i c j c ,c and V ( y [ i c ,j c ] ) c = 2 ∞ X j =1 j = ic j = j c ∞ X i = j +1 i = ic i = j c P ij,c y i y j Due to (3.7) and the assumption P i c j c ,c > V (cid:0) x ( i c ,j c ) (cid:1) c · V ( y [ i c ,j c ] ) c = 0 whence P ij,c = 0 , ∀ i, j ∈ N \{ i c , j c } (3.8)Moreover, we are interested to find the following coefficients P ii c ,c , P ij c ,c for all i ∈ N \{ i c , j c } Furthermore, we assume, there exists i c such that P i c j,c > j = i c or j = j c (herelet j = i c ) (if it is not the case, then V ( x ) c = P i c j c ,c x i c y j c which gives (i)). Without the loss ofgenerality, we may consider i c < i c . Next, let us choose x ( i c ,i c ) = , . . . , , |{z} i thc , , . . . , , |{z} i thc , y [ i c ,i c ] = (cid:0) y , . . . , y i c − , , y i c , . . . , y i c − , , y i c +1 , . . . (cid:1) Using the facts from (3.6) and (3.8), one finds V (cid:0) x ( i c ,j c ) (cid:1) c = 12 P i c i c ,c and V ( y [ i c ,i c ] ) c = 2 ∞ X i =1 i = ic i = ic i = j c P ij c ,c y i y j c Hence, by the same argument as before P ij c ,c = 0 for any i ∈ N \{ i c , i c } . Here, we considertwo subcases: Case 1.
Let P i c j c ,c >
0. By the same argument as before and choosing x ( i c ,j c ) = , . . . , , |{z} i thc , , . . . , , |{z} j thc , y [ i c ,j c ] = (cid:0) y , . . . , y i c − , , y i c , . . . , y j c − , , y j c +1 , . . . (cid:1) we obtain P ii c ,c = 0 for any i ∈ N \{ i c , j c } . Therefore, in this case, we can write V ( x ) c in theform as given by (1). Case 2.
In this case, we suppose that P i c j c ,c = 0. Then, it is clear that we find (2).Now let us turn to ”only if” part. This part comes directly from the fact x ⊥ y , i.e. x k · y k = 0for all k ∈ N . The orthogonality of x and y implies that, for any fixed k ∈ N , either x k = 0 or y k = 0. Therefore, if m ∈ supp ( F π ( k ) ) , k ∈ N , then from (a) one finds V ( x ) m · V ( y ) m = 0.Using (b) one can check that we have V ( x ) c · V ( y ) c = 0 for all c ∈ C F This completes the proof. (cid:3)
We point out that if F = { F k } is an orthogonal system, then for any injective mapping π : N → N , the system F π = { F π ( k ) } is also orthogonal. Hence, the previous theorem will stillremain valid for { F π ( k ) } . Corollary 3.5.
Let F = { F k } be an orthogonal system and V be a QSO such that V ( e k ) = F π ( k ) , k ∈ N , for some injective mapping π : N → N Then, V ( x ) is an OP QSO if and only ifit has the following form, for any x ∈ S : (a) For any m ∈ supp ( F π ( k ) ) V ( x ) m = x k f π ( k ) ,m + ∞ X i =1 a ( m ) ik x i ! where a ( m ) ik = 2 P ik,m − f π ( k ) ,m and set a ( m ) kk = 0 . (b) For any c ∈ C F π , V ( x ) c takes one of the following form (I) If there is no P ij,c > for every i, j ∈ N , then V ( x ) c = 0 or (II) If there exist at least one P i c j c ,c > , then V ( x ) c has one of the following form: (i) If there is no P ij,c > for j ∈ { i c , j c } where i ∈ N \{ j } , then V ( x ) c = 2 P i c j c ,c x i c x j c (ii) If there exist P i c j,c > for either j = i c or j = j c (here let j = i c ), then V ( x ) c has one of the following form: (1) If P i c j c ,c > then V ( x ) c = 2 (cid:0) P i c j c ,c x i c x j c + P i c j c ,c x i c x j c + P i c i c ,c x i c x i c (cid:1) NFINITENESS OF ORTHOGONAL PRESERVING QSO 11 (2) If P i c j c ,c = 0 then V ( x ) c = 2 x i c P i c j c ,c x j c + ∞ X i =1 i = i c ,j c P ii c ,c x i An immediate consequence of the theorem is the following result.
Corollary 3.6.
Let F = { F k } be an orthogonal system and V be a QSO such that V ( e k ) = F π ( k ) , k ∈ N , for some injective mapping π : N → N . Then V ( x ) is an OP QSO if and only ifthe heredity coefficients P ij,k satisfy the following ones: (a) P ii,k = f π ( i ) ,k for k ∈ supp ( F π ( i ) ) , and P ij,k = 0 for k / ∈ { supp ( F π ( i ) ) ∪ supp ( F π ( j ) ) } (b) The coefficients P ij,c , where c ∈ C F π , satisfy one of the following ones: (I) P ij,c = 0 for all i, j ∈ N or (II) If there exist P i c j c ,c > , then P ij,c = 0 for any i, j ∈ N \{ i c , j c } . Further, the othercoefficients must satisfy one of the following, (i) P ij,c = 0 for j ∈ { i c , j c } for all i ∈ N \{ j } or (ii) If there exist P i c j,c > for either j = i c or j = j c (here we let j = i c ), then P ij c ,c = 0 for any i ∈ N \{ i c , j c , i c } . Moreover one of the following must besatisfied: (1) P i c j c ,c > , then P ii c ,c = 0 for any i ∈ N \{ i c , j c , i c } or (2) P i c j c ,c = 0 Corollary 3.7.
Let F = { F k } be an orthogonal system and T is a LSO on S such that T ( e k ) = F π ( k ) for any k ∈ N and an injective mapping π : N → N , then T is an OP linearstochastic operator if and only if T takes the following form: (i) For any m ∈ supp ( F π ( k ) ) T ( x ) m = f k,m x k (ii) For any c ∈ C F π T ( x ) c = 0 Remark 3.8.
Let F = { F k } be an orthogonal system. One of the important class of infinitedimensional OP QSO is when the union of the supports of { F k } cover N . So, let V be a QSOsuch that V ( e k ) = F π ( k ) for some injective mapping π : N → N , and C F π = ∅ . Then V is OP ifand only if one has (i) V has the form given by V ( x ) m = x k f π ( k ) ,m + ∞ X i =1 a ( m ) ik x i ! for any m ∈ supp ( F π ( k ) ) . (ii) The heredity coefficients P ij,k satisfy P ii,k = f i,k ∀ k ∈ supp ( F π ( i ) ) and P ij,k = 0 ∀ k / ∈ { supp ( F π ( i ) ) ∪ supp ( F π ( j ) ) } Now, it is natural to consider an orthogonal system F = { F k } of S such that the support ofeach (or some) F k is countable. Let us provide an example of such kind of orthogonal system. Take A j = { j n : n ≥ } , j ∈ N −
1. It is clear that { A j } is a cover for N . Now, for each j ∈ N − F j = ( f ( j ) m ) ∞ m =1 as follows: for each j ∈ N − f ( j ) m = ( j − j (cid:0) j (cid:1) n , m = j n , n ≥ , , m / ∈ A j . One can see that the system { F j } i ∈ N − is orthogonal and supp ( F j ) = A j , j ∈ N − S . Example 3.9.
Now we are going to produce an example of quadratic shift operator. Assumethat a QSO V such that V ( e i ) = e i +1 for every i ∈ N . From Corollary 3.6 one gets P ii,i +1 = 1 for any i ∈ N . Choose P i , = 0 for any i ≥ . Next, we take for any k ≥ P ik,k +1 = 1 for i ∈ { , , . . . , k − } and P ik,k +1 = 0 for i ≥ k + 1 From the selected heredity coefficients, we have P ij, = 0 for any i, j ∈ N and it is clear thatthey satisfy (2.2) hence V is well-defined. Thus, using Theorem (3.5) one gets ( V ( x )) k = , k = 1 ,x , k = 2 ,x k ∞ P i =1 ,i = k (2 P ik,k +1 − x i ! , k ≥ , k = 1 ,x , k = 2 ,x k (cid:18) k − P i =1 x i + x k ∀ (cid:19) k ≥ Note that V is a concrete example of nonlinear shift operator. Properties of OP QSO
In this section we are going to investigate some properties of infinite dimensional OP QSO.In what follows, we consider proper subsets of N , i.e. α ⊂ N with α = N . For a given α ⊂ N ,we denote Γ α = { x ∈ S : x i = 0 , ∀ i / ∈ α } , ri Γ α = { x ∈ Γ α : x i > , ∀ i ∈ α } By F ix ( V ) we denote the set of all fixed points of V , i.e. F ix ( V ) = { x ∈ S : V ( x ) = x } . Let F = { F k } be an orthogonal system of S . By V F we denote the set of all OP QSO which aregenerated by the orthogonal system F , i.e. V ∈ V F means V ( e k ) = F k for any k ∈ N .Denote supp ( F ) = ∞ [ k =1 supp ( F k ) Lemma 4.1.
Let F = { F k } be an orthogonal system such that supp ( F ) = N and V ∈ V F .Then for any α ⊂ N one has (i) V (Γ α ) ⊂ Γ α ′ ; (ii) V ( ri Γ α ) ⊂ ri Γ α ′ , NFINITENESS OF ORTHOGONAL PRESERVING QSO 13 where α ′ = [ ℓ ∈ α supp ( F ℓ ) . Proof. (i) Let V ∈ V F , then due to Remark 3.8 V takes the following form(4.1) V ( x ) k = x ℓ f ℓ,k + ∞ X i =1 a ( k ) iℓ x i ! for any k ∈ supp ( F ℓ ), ℓ ∈ N .Now let x = ( x , x , . . . ) ∈ Γ α , then x ℓ = 0 for any ℓ / ∈ α , hence from (4.1) one finds V ( x ) k = 0 , for all k ∈ supp ( F ℓ ) , ℓ / ∈ α, (4.2)this is the assertion (i).Now take x = ( x , x , . . . ) ∈ ri Γ α , then x ℓ > ℓ ∈ α . From (4.1) one gets V ( x ) k = x ℓ f ℓ,k + ∞ X i =1 a ( k ) iℓ x i ! = x ℓ f πℓ,k + ∞ X i =1 i = ℓ (2 P iℓ,k − f ℓ,k ) x i = x ℓ f ℓ,k + ∞ X i =1 i = ℓ P iℓ,k x i − f ℓ,k ∞ X i =1 i = ℓ x i = x ℓ f ℓ,k + ∞ X i =1 i = ℓ P iℓ,k x i − f ℓ,k (1 − x ℓ ) = x ℓ X i ∈ α P iℓ,k x i + f ℓ,k x ℓ ! ≥ f ℓ,k x ℓ > k ∈ supp ( F ℓ ). This means V ( ri Γ α ) ⊂ ri Γ α ′ . Moreover, using (4.3) we have supp ( V ( x )) = [ ℓ ∈ α supp ( F ℓ ) . This completes the proof. (cid:3)
Now it is natural to consider the case supp ( F ) ⊂ N . According to Theorem 3.5, for any c ∈ C F (here as before, C = N \ supp ( F )), V ( x ) c takes one of the following form ( i ) V ( x ) c = 0( ii ) V ( x ) c = 2 P i c j c ,c x i c x j c ( iii ) V ( x ) c = 2 (cid:0) P i c j c ,c x i c x j c + P i c j c ,c x i c x j c + P i c i c ,c x i c x i c (cid:1) ( iv ) V ( x ) c = 2 x i c P i c j c ,c x j c + ∞ P i =1 i = i c ,j c P ii c ,c x i (4.4)From now on, let us keep the notation that we have used in Theorem 3.5 (i.e., i c , j c , i c ). To getan analogous result like in Lemma 4.1, it is enough for us to study the coordinates belonging to C F while V ( x ) c takes one of the forms given by (ii), (iii) and (iv), since the case m ∈ supp ( F k )is already described by Lemma 4.1.Let us take α ⊂ N . Now we consider the mentioned cases one by one. CASE (ii) . In this case, we have the following possibilities:( I ) i c , j c ∈ α ; ( II ) i c ∈ α, j c / ∈ α ; ( III ) j c ∈ α, i c / ∈ α ; ( IV ) i c , j c / ∈ α. CASE (iii) . In this case, we have the following ones:( I ) i c , j c , i c ∈ α ; ( II ) i c ∈ α j c , i c / ∈ α ; ( III ) i c , j c ∈ α i c / ∈ α ; ( IV ) i c , i c ∈ α j c / ∈ α. ( V ) j c ∈ α i c , i c / ∈ α ; ( V I ) j c , i c ∈ α i c / ∈ α ; ( V II ) i c , j c , i c / ∈ α ; ( V III ) i c ∈ α j c , i c / ∈ α ; CASE (iv) . This case is the same like CASE (ii).
Remark 4.2.
Let V ∈ V F such that supp ( F ) ⊂ N . For any α ⊂ N we have the followingstatements: (a) Let c ∈ C F , then V ( x ) c takes the form as given by (ii). If (I) is satisfied then V ( x ) c > and in the other cases V ( x ) c = 0 . (b) Let c ∈ C F then V ( x ) c takes the form as given by (iii). If (I), (III), (IV) and (VI) aresatisfied then V ( x ) c > and in the other cases V ( x ) c = 0 . (c) Let c ∈ C F then V ( x ) c takes the form as given by (iv). If - (I) is satisfied then V ( x ) c > (II) is satisfied and there exist i ∈ α such that P i i c ,c > (if not, then V ( x ) c = 0 ),then V ( x ) c > In the other cases V ( x ) c = 0 . Let V be a OP QSO generated by an orthogonal system F = { F k } , i.e. V ( e k ) = F k , k ∈ N .Now want to distinguish a set where some of elements of the system F coincides with certainelements of the standard basis. Namely, let us denote β = { k ∈ N : F k = e i for some i ∈ N } Theorem 4.3.
Let V ∈ V F . If β = ∅ , then for any α ⊂ N , one has F ix ( V ) / ∈ Γ α . Moreover, if the fixed point exists, then
F ix ( V ) ∈ riS . NFINITENESS OF ORTHOGONAL PRESERVING QSO 15
Proof.
Assume that for a fixed point x ∈ S one has x ∈ Γ α for some α ⊂ N . This means V ( x ) k = 0 if k / ∈ α,V ( x ) k > if k ∈ α. Now we consider two separate cases: ( supp ( F ) = N ) and ( supp ( F ) ⊂ N ). Case 1 . Let us suppose ( supp ( F ) = N ). Since x is a fixed point, then one has supp ( V ( x )) = α (4.5)On the other hands, due to the assumption β = ∅ and from Lemma 4.1, we get | supp ( F k ) | ≥ k ∈ α and supp ( V ( x )) = [ ℓ ∈ α supp ( F ℓ )Therefore | supp ( V ( x )) | > | α | (4.6)which contradicts to (4.5). Therefore, the fixed point cannot be in the face Γ α for any α ⊂ N . Part 2 ( supp ( F ) ⊂ N ) . Take any α ⊂ N . Now we are going to consider the following threepossible cases: C F ∩ α = ∅ , C F ∩ α = ∅ , α
6⊂ C F , and α ⊆ C F .In the first case, we obtain the desired result by the same argument as in Part 1 .Now we consider the case: α \C F = ∅ . Let x ∈ ri Γ α which implies (4.5). On the other hand,we have | supp ( F π ( k ) ) | ≥ k ∈ α \C F , therefore using Lemma 4.1 one concludes that | supp ( V ( x )) | = |{ α ∩ C F } ∪ [ k ∈ α \C F supp ( F π ( k ) ) | > | a | which contradicts to (4.5).Let us turn to the last case, i.e. α ⊆ C F . Due to x ∈ ri Γ α we get (4.5) and X k ∈ α V ( x ) k = 1(4.7)On the other hands, by taking into account that x ∈ ri Γ α and P ii,c = 0 for any i ∈ N , c ∈ C F , then one finds X k ∈ α V ( x ) k = X k ∈ α X i,j ∈ α i = j P ij,k x i x j = X i,j ∈ α i = j x i x j X k ∈ α P i c j c ,k ! (4.8)Since P k ∈ N P ij,k = 1, we then obtain X k ∈ α V ( x ) k ≤ X i,j ∈ α i = j x i x j = X i ∈ α x i X j ∈ α j = i x j (4.9)Again x ∈ ri Γ α implies X j ∈ α j = i x j < X j ∈ α x j = 1 for any i ∈ α Therefore, X k ∈ α V ( x ) k ≤ X i,j ∈ α i = j x i x j < X i ∈ α x i = 1(4.10)which contradicts to (4.7).Furthermore, according to the arbitrariness of α ⊂ N , we infer that if a fixed point x exists,then x ∈ riS . This completes the proof. (cid:3) Remark 4.4.
Let F = { F k } be an orthogonal system in S and α ⊂ N . Then we have supp ( { F k } k ∈ α ) = supp ( { e k } k ∈ α ) if and only if there is a permutation π α of α such that { F k } k ∈ α = { e π α ( k ) } k ∈ α . Theorem 4.5.
Let V be an OP QSO generated by V ( e k ) = F k for any k ∈ N and let set β = ∅ .Assume that for any α ⊂ β one has { F k } k ∈ α = { e π α ( k ) } k ∈ α for any permutation π α of α . Then for any α ⊂ N one has F ix ( V ) / ∈ Γ α Moreover, if a fixed point exists, then
F ix ( V ) ∈ riS .Proof. Assume that for a fixed point x ∈ S one has x ∈ Γ α for some α ⊂ N . Without loss ofgenerality we may assume that x ∈ ri Γ α . Now we consider two possibilities supp ( F ) = N and supp ( F ) ⊂ N . Part 1 ( supp ( F ) = N ) . There are several possibilities:(a) α ∩ β = ∅ (b) α ∩ β = ∅ , α β (c) α ⊂ β Cases (a) and (b) follow from the same argument as in the proof of Theorem 4.3, since thereexists some k ∈ α \ β such that supp ( F k ) ≥ α ⊂ β . Due to our assumption, we have supp ( x ) = supp ( V ( x )) = supp ( { e k } k ∈ α ) = α (4.11)From Lemma 4.1 one gets that supp ( V ( x )) = supp ( { F } k ∈ α )(4.12)From (4.11), (4.12) and Remark 4.4 we conclude that there is a permutation π α of α such that { F k } k ∈ α = { e π α ( k ) } k ∈ α . which contradicts to the assumption of the theorem. Part 2 ( supp ( F ) ⊂ N ) . Since we have already considered all possible situations of α and β ,therefore, then it is enough for us to consider the following cases: α ∩C F = ∅ , α ∩C F = ∅ , α
6⊂ C F and α ⊂ C F . These cases can be proceeded by the same argument as in the proof of Theorem4.3. This completes the proof. (cid:3) Now we want provide certain examples which satisfy the conditions of the last theorem.
NFINITENESS OF ORTHOGONAL PRESERVING QSO 17
Example 4.6.
Let us consider the following orthogonal system: F = (cid:18) , , , . . . , (cid:19) , F = e , F = e , F = e , F n = , . . . , , |{z} n − , |{z} n , , . . . , for n ≥ Let V be generated as follows V ( e k ) = F k , k ∈ N . One can see that the set β = { , , } andfor any subset A ⊂ β we have { F k } k ∈ A = { e k } k ∈ A Then, due to Theorem 4.5 for any α ⊂ N , we have F ix ( V ) / ∈ Γ α . Example 4.7.
Let us consider the following orthogonal system: F = (cid:18) , , , . . . , (cid:19) , F = e , F = e , F = e , F n = , . . . , , |{z} n − , |{z} n , , . . . , for n ≥ Let V be generated as follows V ( e k ) = F k , k ∈ N . One can see that β = { , , } and C F = { , , } . Moreover, one has for any subset A ⊂ β { F k } k ∈ A = { e k } k ∈ A Then, due to Theorem 4.5 for any α ⊂ N , we have F ix ( V ) / ∈ Γ α . It is well-known that an infinite-dimensional simplex S is not compact either in ℓ topology,nor in a weak topology, therefore, the existence of a fixed point of any QSO V defined on S isnot always true. Example 4.8.
Let us consider an OP QSO V defined by V ( x , x , · · · , x n , · · · ) = (0 , x , x , · · · , x n , · · · ) where ( x n ) ∈ S . It is easy to see that this operator has no fixed points belonging to S . Next result provides a sufficient condition for the existence of a fixed point of OP QSO.
Proposition 4.9.
Let V ∈ V F with supp ( F ) = N . If β = ∅ and there exists a subset α ⊆ β with | α | < ∞ such that { e π ( k ) } k ∈ α = { F k } k ∈ α for some permutation π of α . Then there exists a fixed point x ∈ Γ α .Proof. Let α = { i , . . . , i n } ⊆ β . By the definition of V we infer that V ( e i k ) = e π ( i k ) for all k ∈ { , . . . , n } (4.13)and V ( e m ) = F m for all m ∈ N \ α Due to Corollary 3.8 the operator V can be written in the following form, for any x ∈ Γ α V ( x ) π ( i ) = x i P ℓ ∈ α ℓ = i (2 P ℓi,j − x ℓ , i ∈ αV ( x ) k = 0 if k / ∈ α (4.14)This implies that V (Γ α ) ⊂ Γ α . The compactness of Γ α with the Brouwer fixed-point Theoremyields the existence of a fixed point x ∈ Γ α of V . This finishes the proof. (cid:3) Immediately from the last proposition, one concludes the following corollary.
Corollary 4.10.
Let V ∈ V F with supp ( F ) ⊂ N and β = ∅ . If V ( x ) c = 0 for any c ∈ C F andthere exists a subset α ⊆ β with | α | < ∞ such that { e π ( k ) } k ∈ α = { F k } k ∈ α for some permutation π of α . Then there exists a fixed point x ∈ Γ α . We provide an example of OP QSO that has fixed point.
Example 4.11.
Let us consider the following orthogonal system: F = e , F = e , F n = , . . . , , |{z} n − , |{z} n − , , . . . , for n ≥ Now let V be an OP QSO such that V ( e ) = F , V ( e ) = F , V ( e k ) = F k − for k ≥ One can see that β = { , } , and for a permutation π (1) = 2 , π (1) = 2 , we have { e π ( k ) } k ∈ β = { F k } k ∈ β . For any x ∈ ri Γ β , using Corollary 3.8, one gets (cid:26) V ( x ) = x (1 + (2 P , − x ) V ( x ) = x (1 + (2 P , − x )(4.15) In particular, assume that P , = P , = 12 , then clearly we have (cid:18) , (cid:19) as a fixed point forthe system (4.15) . Clearly, (cid:18) , , , . . . , (cid:19) ∈ Γ { , } is a fixed point for V . Acknowledgments
The present work is supported by the UAEU ”Start-Up” Grant, No. 31S259.
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E-mail address : [email protected], [email protected] Ahmad Fadillah, Department of Computational & Theoretical Sciences, Faculty of Science,International Islamic University Malaysia, Kuantan, Pahang, Malaysia
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