Submajorization on \ell^p(I)^+ determined by proper doubly substochastic operators and its linear preservers
aa r X i v : . [ m a t h . F A ] F e b Submajorization on ℓ p ( I ) + determined by properdoubly substochastic operators and its linearpreservers Martin Z. Ljubenovi´c, Dragan S. Raki´c ∗ Abstract
We note that the well-known result of Von Neumann [29] is notvalid for all doubly substochastic operators on discrete Lebesgue spaces ℓ p ( I ), p ∈ [1 , ∞ ). This fact lead us to distinguish two classes of theseoperators. Precisely, the class of proper doubly substochastic oper-ators on ℓ p ( I ) is isolated with the property that an analogue of theVon Neumann result on operators in this class is true. The subma-jorization relation ≺ s on the positive cone ℓ p ( I ) + , when p ∈ [1 , ∞ ),is introduced by proper substochastic operator and it is provided thatsubmajorization may be considered as a partial order. Two differentshapes of linear preservers of submajorization ≺ s on ℓ ( I ) + and on ℓ p ( I ) + , when I is an infinite set, are presented. Key words and phrases : majorization, submajorization, stochasticoperators, permutation, preservers.
Mathematics subject classification : 47B60, 15B51, 39B62, 60E15.
A square n × n non-negative matrix A = ( a ij ) is called a doubly stochasticif all of its row sums and all of its column sums are equal 1. A square n × n matrix D = ( d ij ) with non-negative entries is called doubly substochastic ifthere is a doubly stochastic matrix A = ( a ij ) such that d ij ≤ a ij , ≤ i ≤ n, ≤ j ≤ n. (1) ∗ This research was financially supported by the Ministry of Education, Science andTechnological Development of the Republic of Serbia and the bilateral project betweenSerbia and Slovenia (Generalized inverses, operator equations and applications) Grant No.337-00-21/2020-09/32
1s a direct consequence, it follows that all row sums and column sumsof matrix D are less than or equal to 1. Von Neumann [29] provides theconverse in the next theorem. Theorem 1.1. [26, Theorem I.2.C.1] For every n × n non-negative matrix D = ( d ij ) with n X i =1 d ij ≤ , ∀ j ∈ { , , . . . , n } and n X j =1 d ij ≤ , ∀ i ∈ { , , . . . , n } , (2) there exists a doubly stochastic matrix A = ( a ij ) such that d ij ≤ a ij , for all i, j ∈ { , , . . . , n } . Thus, the statement (2) may be considered as an alternative definitionof a doubly substochastic matrix. However, the last result of von Neumannmay not be generalized to infinite matrices considered as bounded linearoperators on discrete Lebesgue spaces ℓ p ( I ), whenever p ∈ [1 , ∞ ). Definition 1.1. [19, Definition 3.1] Let p ∈ [1 , ∞ ) and let A : ℓ p ( J ) −→ ℓ p ( I ) be a bounded linear operator, where I and J are two non-empty sets.The operator A is called • row substochastic , if A is positive and ∀ i ∈ I , P j ∈ J h Ae j , e i i ≤ • column substochastic , if A is positive and ∀ j ∈ J , P i ∈ I h Ae j , e i i ≤ • doubly substochastic , if A is both row and column substochastic. Definition 1.2. [6, Definition 2.1] Let p ∈ [1 , ∞ ) and let A : ℓ p ( J ) −→ ℓ p ( I )be a positive bounded linear operator, where I and J are two non-emptysets. The operator A is called doubly stochastic , if ∀ i ∈ I, X j ∈ J h Ae j , e i i = 1 , and ∀ j ∈ J, X i ∈ I h Ae j , e i i = 1 . It is easy to see that the left and the right shift operators
L, R : ℓ p ( N ) → ℓ p ( N ) defined by h Le j , e i i := (cid:26) , j − i = 1 , , otherwise h Re j , e i i := (cid:26) , i − j = 1 , , otherwise2ith matrix forms L = . . . . . . . . . . . . . . . ... ... ... ... . . . R = . . . . . . . . . . . . . . . ... ... ... ... . . . (3)are doubly substochastic, but there is no doubly stochastic operator A suchthat h Le j , e i i ≤ h Ae j , e i i , ∀ i, j ∈ N or h Re j , e i i ≤ h Ae j , e i i , ∀ i, j ∈ N holds (the definition (3) of the matrix operators L and R is supported byTheorem 2.1). We conclude that shift operators L and R are not doublysubstochastic in the sense of definition (1). This fact lead us to considera subclass of doubly substochastic operators which is called proper doublysubstochastic operators (introduced in Definition 3.1) based on the definition(1) to get that each operator from this class satisfies conditions (1) and (2)(reformulated for operators on ℓ p ( I )).In recent years, there is a big progress towards developing various ex-tensions of the most important majorization relations on sequence spaces[5, 15] and on descrete Lebesgue spaces [6, 7, 8, 9, 10, 18, 19, 20, 21, 22, 23]with apropriate generalizations of some famous theorems in linear algebra[2, 3, 4, 16, 24, 27, 30, 32]. There are a lot of applications of the majorizationtheory in various branches of mathematics and there exist significant conec-tions with the other science like physics, quantum mechanics and quantuminformation theory [12, 17, 25, 30, 31, 33].We recommend clasical monographs [11, 13, 26] as collections of themost important inequalities and results in the finite-dimensional majoriza-tion theory with their applications onto various fields in mathematics. Seealso [1, 14, 28, 29].Notations, preliminaries and some published results which are used inthis paper are contained in Section 2. Section 3 provides some useful prop-erties of introduced proper doubly stochastic operators and submajorizationrelation on ℓ p ( I ) + , defined by f ≺ s g whenever f = Dg for some proper dou-bly substochastic operator D . This relation represents an extension of thesubmajorization between two n dimensional vectors [26, Theorem I.2.C.4]3hich contains positive real numbers. We provide that, in some sense, thesubmajorization relation may be considered as a partial order on the positivecone ℓ p ( I ) + , for some p ∈ [1 , ∞ ).In Section 4 we analyze bounded linear operators on ℓ p ( I ) which preservesubmajorization relation ≺ s . Actually, we split the problem on two cases,the submajorization on ℓ p ( I ) + when p ∈ (1 , ∞ ) and the submajorization on ℓ ( I ) + . Some complex and technically demanding lemmas and theorems areproved in order to provide and present concrete forms of linear preservers ofsubmajorization ≺ s (Corollaries 4.1 and 4.2). In this section we will present notations and the most important resultswhich we will use in the paper.Let f : I −→ R be an arbitrary function, where I is a non-empty set.The function f is summable if there exists a real number σ with the followingproperty:For every ǫ > J ⊆ I such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ − X j ∈ J f ( j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ whenever J is a finite set and J ⊆ J . In this case, σ is called the sum of f and we use denotation σ = P i ∈ I f ( i ).The Banach space of all functions f : I −→ R , where I is non-empty setand p ∈ [1 , ∞ ) with the property P i ∈ I | f ( i ) | p < ∞ is considered, and it isdenoted by ℓ p ( I ). This space is equipped with p - norm k f k p := X i ∈ I | f ( i ) | p ! p < ∞ . Each function f ∈ ℓ p ( I ) may be represented by f = P i ∈ I f ( i ) e i , wherefunctions e i : I −→ R , i ∈ I are defined by Kronecker delta, i.e., e i ( j ) = δ ij ,for all j ∈ I . We will consider positive elements of the Banach space ℓ p ( I ), p ∈ [1 , ∞ ), and denote this positive cone by ℓ p ( I ) + := { f ∈ ℓ p ( I ) : f ( i ) ≥ , ∀ i ∈ I } . In some situations we will consider I + f as a subset of I determined by I + f := { i ∈ I : f ( i ) > } , f ∈ ℓ p ( I ).A q is conjugate (dual) exponent of p if p + q = 1, when p, q ∈ (1 , ∞ ).Moreover, the exponents 1 and ∞ are considered to be dual exponent toeach other. For all g ∈ ℓ q ( I ), the map f −→ h f, g i := P i ∈ I f ( i ) g ( i ) definesa bounded linear functional on ℓ p ( I ). Since ℓ p ( I ) ∗ is isometrically isomorphicwith ℓ q ( I ), the dual Banach space ℓ p ( I ) ∗ can be identified with ℓ q ( I ). Themap h· , ·i : ℓ p ( I ) × ℓ q ( I ) −→ R defined by h f, g i = P i ∈ I f ( i ) g ( i ) is calledthe dual pairing. Using the dual pairing h· , ·i for functions f ∈ ℓ p ( I ) and e i ∈ ℓ ( I ) ⊂ ℓ q ( I ), we have the representation f ( i ) = h f, e i i , ∀ i ∈ I, and f = X i ∈ I h f, e i i e i . A bounded linear operator A : ℓ p ( I ) → ℓ p ( I ) may be represented by amatrix [ a ij ] i,j ∈ I , which may be finite or infinite depends on cardinality ofthe set I . If we define matrix elements with a ij := h Ae j , e i i , ∀ i, j ∈ I we getthe matrix representation of operator A in the following way Af ( i ) = X j ∈ I a ij f ( j ) , ∀ i ∈ I, (4)that is, Af = X i ∈ I X j ∈ I a ij f ( j ) e i . Theorem 2.1. [22, Corollary 3.1] Let A = { a ij : i, j ∈ I } be a family ofreal numbers. If this family satisfies conditions sup j ∈ I X i ∈ I | a ij | < ∞ (5) and sup i ∈ I X j ∈ I | a ij | < ∞ (6) then this family may be considered as an bounded linear operator A on ℓ p ( I ) ,for every p ∈ [1 , ∞ ] , defined by Af := X i ∈ I X j ∈ I a ij f ( j ) e i . (7)5et p ∈ [1 , ∞ ) and A : ℓ p ( I ) −→ ℓ p ( I ) be a bounded linear operator,where I is a non-empty set. The operator A is called • a permutation , if there exists a bijection θ : I −→ I for which Ae j = e θ ( j ) , for each j ∈ I ; • a partial permutation for sets I ⊂ I and I ⊂ I , if there exist abijection θ : I −→ I for which Ae j = e θ ( j ) whenever j ∈ I , and Ae j = 0, otherwise.The set of all permutations and set of all partial permutations on ℓ p ( I ) aredenoted by P ( ℓ p ( I )) and pP ( ℓ p ( I )), respectively. Row, column and doublysubstochastic operators on ℓ p ( I ) presented in Definition 1.1 are denoted by RsS ( ℓ p ( I )), CsS ( ℓ p ( I )) and DsS ( ℓ p ( I )), respectively. Doubly stochasticoperators form Definition 1.2 we will denote by DS ( ℓ p ( I )). Definition 2.1. [19, Definition 4.1] Let p ∈ [1 , ∞ ). For two functions f, g ∈ ℓ p ( I ) + , the function f is weakly majorized by g , if there exists adoubly substochastic operator D ∈ DsS ( ℓ p ( I )) such that f = Dg , anddenote it by f ≺ w g . Theorem 2.2. [19, Theorem 4.2] Let f, g ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) . The nextstatements are equivalent: • f ≺ w g and g ≺ w f ; • There exist P ∈ pP ( ℓ p ( I )) for sets I + f and I + g such that g = P f . Definition 2.2. [6, Definition 3.1] Let p ∈ [1 , ∞ ). For two functions f, g ∈ ℓ p ( I ), the function f is majorized by g , if there exists a doubly stochasticoperator D ∈ DS ( ℓ p ( I )) such that f = Dg , and denote it by f ≺ g . Theorem 2.3. [19, Theorem 4.3] Let f, g ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) . The nextstatements are equivalent: • f ≺ w g and g ≺ f ; • There exist P ∈ P ( ℓ p ( I )) such that g = P f . The space ℓ p ( I ) is an ordered Banach space under the natural partialordering on the set of real valued functions defined on I . The operator A : ℓ p ( I ) −→ ℓ p ( I ), p ∈ [1 , ∞ ) is called positive if Ag ∈ ℓ p ( I ) + for every g ∈ ℓ p ( I ) + .An operator A ∗ : ℓ q ( I ) −→ ℓ q ( I ) is the adjoint operator of A : ℓ p ( I ) −→ ℓ p ( I ), p ∈ [1 , ∞ ), if h Af, g i = h f, A ∗ g i , ∀ f ∈ ℓ p ( I ), ∀ g ∈ ℓ q ( I ), where q isthe conjugate exponent of p . 6 efinition 2.3. [20, 21] Let p ∈ [1 , ∞ ). A bounded linear operator T : ℓ p ( I ) → ℓ p ( I ) is called a preserver of weak majorization on ℓ p ( I ) + , if T preserves the weak majorization relation, that is, T f ≺ w T g , whenever f ≺ w g , where f, g ∈ ℓ p ( I ) + . The set of all linear preservers of weakmajorization on ℓ p ( I ) + is denoted by P w ( ℓ p ( I ) + ).Let θ : I → I be one-to-one function. Let P θ : ℓ p ( I ) → ℓ p ( I ), p ∈ [1 , ∞ )be a bounded linear operator defined in the following way P θ ( f ) := X k ∈ I f ( k ) e θ ( k ) , f ∈ ℓ p ( I ) . (8) Theorem 2.4. [20, Theorem 3.5] Let p ∈ (1 , ∞ ) , and let I be an infinite set.Suppose that T : ℓ p ( I ) → ℓ p ( I ) is a bounded linear operator. The followingstatements are equivalent:i) T ∈ P w ( ℓ p ( I ) + ) .ii) T e j ≺ w T e k and T e k ≺ w T e j , ∀ k, j ∈ I , and for each i ∈ I there is atmost one j ∈ I such that h T e j , e i i > .iii) T = P k ∈ I λ k P θ k , where ( λ k ) k ∈ I ∈ ℓ p ( I ) + , I ⊂ I is at most countable, θ k ∈ Θ := { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } . (9)We use two classes of operators on ℓ ( I ). Let P ( ℓ ( I ) + ) be the set ofall bounded linear operators on ℓ ( I ) + defined by T = X k ∈ I λ k P θ k , (10)where I is at most a countable subset of I , ( λ k ) k ∈ I ∈ ℓ ( I ) + , every θ k belongs to a countable family of one-to-one maps with disjont ranges as in(9). We denote by P ( ℓ ( I ) + ), the set of all bounded linear operators definedby T h ( f ) := h X i ∈ I f ( i ) , ∀ f ∈ ℓ ( I ) , (11)where h ∈ ℓ ( I ) + . It is easy to see that each ”row” of the operator T h ,for example k ∈ I , contains the same elements h ( k ). In the other words,all ”columns” of the operator T h are the same and they are equal withfunction h . These two classes of operators are introduced in papers [6, 21]and operators in these classes preserve the weak majorization on ℓ ( I ) + which is provided in the next result. 7 heorem 2.5. [21, Theorem 3.3] Let T : ℓ ( I ) → ℓ ( I ) be a bounded linearoperator, where I is an infinite set. The following statements are equivalent:(i) T ∈ P w ( ℓ ( I ) + ) ;(ii) There are operators T ∈ P ( ℓ ( I ) + ) and T ∈ P ( ℓ ( I ) + ) and disjointsets I , I ⊂ I with I ∪ I = I such that T = T + T where T , T arechosen to be h T f, e i i = h T f, e i i = 0 , ∀ i ∈ I , ∀ i ∈ I , ∀ f ∈ ℓ ( I ) + ; (iii) There is an at most a countable set I ⊂ I and there is a family Θ := { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } of one-to-one maps, θ k ∈ Θ , ∀ k ∈ I , and ( λ i ) i ∈ I ∈ ℓ ( I ) + such that T = X k ∈ I λ k P θ k + T h , where T h ( f ) := h P k ∈ I f ( k ) , for h ∈ ℓ ( I ) + with h h, e j i = 0 , ∀ j ∈ S i ∈ I θ i ( I ) ;(iv) T e j ≺ w T e k and T e k ≺ w T e j , ∀ k, j ∈ I , and for each i ∈ I , eitherthere exists exactly one j ∈ I with h T e j , e i i > or the set {h T e j , e i i| j ∈ I } is a singleton. Lemma 2.1. [21, Lemma 3.1] Let u = { u j } ∈ R n and let { u ij | i ∈ I , j =1 , . . . , n } be a family of real numbers, where I is at most a countable set. If n X j =1 α j u j ∈ n X j =1 α j u ij | i ∈ I , (12) for all α = ( α , α , . . . , α n ) with α j > for each j = 1 , . . . , n , then thereexists k ∈ I such that u j = u kj , for each j = 1 , . . . , n . At the beginning of this section, we will introduce the notion of properdoubly substochastic operators, based on the relation (1) for the finite-dimensional case. 8 efinition 3.1.
Let p ∈ [1 , ∞ ) and let A : ℓ p ( I ) −→ ℓ p ( I ) be a positivebounded linear operator, where I is a non-empty set. The operator A iscalled proper doubly substochastic , if there is doubly stochastic operator A : ℓ p ( I ) −→ ℓ p ( I ) such that ∀ i ∈ I, ∀ j ∈ I, h Ae j , e i i ≤ h A e j , e i i . (13) Remark 3.1.
For a doubly stochastic operator A : ℓ p ( J ) −→ ℓ p ( I ) have tobe card ( I ) = card ( J ), by [6, Theorem 2.2]. We could define in Definition 3.1proper doubly substochastic operators from ℓ p ( J ) to ℓ p ( I ) as in Definition1.1. However, since the Definition 3.1 of the proper doubly substochasticoperators uses inequalities (13), we conclude that have to be card ( I ) = card ( J ), so we may set I = J in Definition 3.1.Proper doubly substochastic operators on ℓ p ( I ) introduced in the aboveDefinition 3.1 will be denoted by pDsS ( ℓ p ( I )). It is easy to see that pDsS ( ℓ p ( I )) ( DsS ( ℓ p ( I )) = RsS ( ℓ p ( I )) ∩ CsS ( ℓ p ( I )) . Furthermore, the norm of doubly substochastic operators is less then orequal to 1, by [19, Lemma 3.3]. Using Definition 3.1, the next lemma isstraightforward.
Lemma 3.1.
Let p ∈ [1 , ∞ ) . For each proper doubly substochastic oper-ator D ∈ pDsS ( ℓ p ( I )) there are two operators D ∈ DS ( ℓ p ( I )) and D ∈ DsS ( ℓ p ( I )) such that D = D + D . Lemma 3.2.
Let p ∈ [1 , ∞ ) . The set pDsS ( ℓ p ( I )) is closed under thecomposition.Proof. Let
A, B ∈ pDsS ( ℓ p ( I )). Clearly, AB is a positive operator. UsingDefinition 3.1 there are corresponding operators A , B ∈ DS ( ℓ p ( I )) suchthat ∀ i ∈ I, ∀ j ∈ I, h Ae j , e i i ≤ h A e j , e i i and h Be j , e i i ≤ h B e j , e i i . Now, for arbitrary chosen i, k ∈ I we get h ABe k , e i i = h A ( Be k ) , e i i = A ( Be k )( i ) = X j ∈ I h Ae j , e i ih Be k , e j i≤ X j ∈ I h A e j , e i ih B e k , e j i = A ( B e k )( i )= h A B e k , e i i . (14)9ince DS ( ℓ p ( I )) is closed under the composition by [6, Theorem 2.4] weobtain A B ∈ DS ( ℓ p ( I )), so AB ∈ pDsS ( ℓ p ( I )), by (14). Theorem 3.1.
Let p ∈ [1 , ∞ ) . The set pDsS ( ℓ p ( I )) is convex.Proof. Let
A, B ∈ pDsS ( ℓ p ( I )) and suppose that A , B ∈ DS ( ℓ p ( I )) suchthat h Ae j , e i i ≤ h A e j , e i i and h Be j , e i i ≤ h B e j , e i i , for all i, j ∈ I . Let C = tA + (1 − t ) B . Clearly, C is a positive operator. Furthermore, h Ce j , e i i = t h Ae j , e i i + (1 − t ) h Be j , e i i≤ t h A e j , e i i + (1 − t ) h B e j , e i i = h C e j , e i i , ∀ i, j ∈ I, where C = tA + (1 − t ) B . It is easy to see that C ∈ DS ( ℓ p ( I )) because DS ( ℓ p ( I )) is a convex set, by [18, Theorem 3.3]. Thus, C ∈ pDsS ( ℓ p ( I )),so it follows that pDsS ( ℓ p ( I )) is a convex set. Definition 3.2.
Let p ∈ [1 , ∞ ). For two functions f, g ∈ ℓ p ( I ) + , the func-tion f is submajorized by g , if there exists a proper doubly substochasticoperator D ∈ pDsS ( ℓ p ( I )) such that f = Dg , which is denoted by f ≺ s g .Clearly, f ≺ s g implies f ≺ w g , by pDsS ( ℓ p ( I )) ⊂ DsS ( ℓ p ( I )). Theopposite direction is not true in general. Example 3.1.
Let p ∈ [1 , ∞ ), g = ( g , g , g , . . . , g n , . . . ) ∈ ℓ p ( N ), where g i > ∀ i ∈ N and g i < g j , whenever i > j . Let f := Rg = (0 , g , g , g , . . . , g n , . . . ) , where R is the right shift operator defined in (3). Clearly f ≺ w g , since R ∈ DsS ( ℓ p ( I )). We claim that f s g . Suppose that f = Dg holds for anarbitrary chosen doubly substochastic operator D : ℓ p ( N ) → ℓ p ( N ). Now,0 = f = ∞ X j =1 h De j , e i g j so we get h De j , e i = 0, ∀ j ∈ N , because g is a strictly decreasing sequencewith non-zero elements. Further, g = f = ∞ X j =1 h De j , e i g j ≤ ∞ X j =1 h De j , e i g ≤ g ,
10o the last inequality holds only when h De , e i = 1 and h De j , e i = 0whenever j = 1 . Because D ∈ DsS ( ℓ p ( I )) ⊂ CsS ( ℓ p ( I )) it follows that h De , e i i = 0, whenever i = 2. Now, g = f = ∞ X j =1 h De j , e i g j = 0 + ∞ X j =2 h De j , e i g j ≤ ∞ X j =2 h De j , e i g ≤ g . Similarly as above, we get that h De , e i = 1 and h De j , e i = 0 whenever j = 1, and since D ∈ CsS ( ℓ p ( I )) it follows that h De , e i i = 0, whenever i =3. Continuing this process we obtain that have to be D = R pDsS ( ℓ p ( I )).Thus f s g . Lemma 3.3.
Let p ∈ [1 , ∞ ) and let f, g ∈ ℓ p ( I ) + . If f ≺ s g and g ≺ w f ,then f ≺ g .Proof. Let { I nf : n ∈ N } be a family of disjoint finite subsets of I + f forarbitrary chosen f ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) defined by I f := n i ∈ I + f : f ( i ) = max { f ( j ) : j ∈ I + f } o and I nf := ( i ∈ I + f : f ( i ) = max ( f ( j ) : j ∈ I + f \ n − [ k =1 I kf )) whenever n ≥
2. Above maximums exist by the definition of ℓ p ( I ) + . Obvi-ously, I + f = S ∞ k =1 I kf . If I kf = ∅ , for some k ∈ N then we define f k := f ( j ),for some j ∈ I kf . Otherwise f k := 0.Let f ≺ s g and g ≺ w f . Since f ≺ s g implies f ≺ w g , there exists P ∈ pP ( ℓ p ( I )) for sets I + f and I + g such that f = P g , by Theorem 2.2.Because of this, it is easy to see that f i = g i and card ( I if ) = card ( I ig ) , ∀ i ∈ N . (15)Since f ≺ s g , there is D ∈ pDsS ( ℓ p ( I )) such that f = Dg so usingLemma 3.1 there exist two operators D ∈ DS ( ℓ p ( I )) and D ∈ DsS ( ℓ p ( I ))such that D = D + D . We claim that D g = 0 . (16)11et i ∈ I f . Then f = f ( i ) = X j ∈ I g ( j ) De j ( i )= X j ∈ I g g De j ( i ) + X j ∈ I \ I g g ( j ) De j ( i ) ≤ X j ∈ I g g De j ( i ) + X j ∈ I \ I g g De j ( i ) ≤ g = f . It follows that P j ∈ I g De j ( i ) = 1 and P j ∈ I \ I g De j ( i ) = 0, by D ∈ pDsS ( ℓ p ( I )).Thus, for every i ∈ I f we obtain D e j ( i ) = De j ( i ) and D e j ( i ) = 0 , ∀ j ∈ I. Since, card ( I g ) = card ( I f ) = X i ∈ I f X j ∈ I g De j ( i ) = X j ∈ I g X i ∈ I f De j ( i ) , we get P i ∈ I f De j ( i ) = 1, ∀ j ∈ I g and P i ∈ I \ I f De j ( i ) = 0, ∀ j ∈ I g . Thus, ∀ j ∈ I g we have D e j ( i ) = De j ( i ) and D e j ( i ) = 0 , ∀ i ∈ I. Let k ∈ I f . Using above facts we obtain f = f ( k ) = X j ∈ I g ( j ) De j ( k )= X j ∈ I g g De j ( k ) + X j ∈ I g g ( j ) De j ( k ) + X j ∈ I \{ I g ∪ I g } g ( j ) De j ( k )= X j ∈ I g g De j ( k ) + 0 + X j ∈ I \{ I g ∪ I g } g ( j ) De j ( k ) ≤ X j ∈ I g g De j ( k ) + X j ∈ I \{ I g ∪ I g } g De j ( k ) ≤ g = f . It follows that P j ∈ I g De j ( k ) = 1 and P j ∈ I \ I g De j ( k ) = 0, by D ∈ pDsS ( ℓ p ( I )).Thus, D e j ( k ) = De j ( k ) and D e j ( k ) = 0 , ∀ j ∈ I, k ∈ I f .Similarly as above, using card ( I g ) = card ( I f ) and changing the order ofsummation, we get card ( I g ) = X j ∈ I g X i ∈ I f De j ( i ) , so P i ∈ I f De j ( i ) = 1, ∀ j ∈ I g and P i ∈ I \ I f De j ( i ) = 0, ∀ j ∈ I g . Thus ∀ j ∈ I g we have D e j ( i ) = De j ( i ) and D e j ( i ) = 0 , ∀ i ∈ I. Continuing this process, we get for arbitrary chosen n ∈ N that D e j ( k ) = 0 , ∀ k ∈ I nf , ∀ j ∈ I. (17)and D e j ( i ) = 0 , ∀ j ∈ I ng , ∀ i ∈ I, (18)hold. Finally, using (18) we obtain D g ( i ) = X j ∈ I g ( j ) D e j ( i ) = X j ∈ I + g g ( j ) D e j ( i ) = ∞ X n =1 X j ∈ I ng g ( j ) D e j ( i ) = 0 , for each i ∈ I . Now, f = Dg = ( D − D ) g = D g , that is f ≺ g .As direct consequence of Lemma 3.3, we get the following three corol-laries. Corollary 3.1.
Let f, g ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) . The next statements areequivalent: • f ≺ s g and g ≺ w f ; • There exist P ∈ P ( ℓ p ( I )) such that g = P f .Proof.
Let f ≺ s g and g ≺ w f . Using Lemma 3.3 we get that f ≺ g . Therest follows by Theorem 2.3.Conversely, if g = P f for P ∈ P ( ℓ p ( I )) ⊂ pDsS ( ℓ p ( I )) it follows that f = P − g where P − ∈ P ( ℓ p ( I )) ⊂ DsS ( ℓ p ( I )), so we get f ≺ s g and g ≺ w f .The next example shows that condition f ≺ s g in the above corollarycan not be replaced by f ≺ w g . 13 xample 3.2. Let f ∈ ℓ ( N ) defined by f ( i ) = 1 /i , i ∈ N that is, f = (1 , , , . . . ). Using right shift operator R , we get that g := Rf =(0 , , , , . . . ). Similarly, f = Lg . It follows that f and g are mutuallyweakly majorized so they are different up to the partial permutation byTheorem 2.2. However, g (1) = 0 f ( I ). Thus there is no permutation P ∈ ℓ ( I ) to be f = P g .Since g ≺ s f implies g ≺ w f , using Corollary 3.1 we obtain the followingresult. Corollary 3.2.
Let f, g ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) . The next statements areequivalent: • f ≺ s g and g ≺ s f ; • There exist P ∈ P ( ℓ p ( I )) such that g = P f . As corollary of the above result we obtain an analogue of [6, Theorem3.5] for positive cone ℓ p ( I ) + . Corollary 3.3.
Let f, g ∈ ℓ p ( I ) + , p ∈ [1 , ∞ ) . The next statements areequivalent: • f ≺ g and g ≺ f ; • There exist P ∈ P ( ℓ p ( I )) such that g = P f . Corollary 3.4.
The submajorization relation ” ≺ s ” when p ∈ [1 , ∞ ) , isreflexive and transitive relation i.e. ” ≺ s ” is a pre-order. If we identify allfunctions which are different up to the permutation, then we may consider” ≺ s ” as a partial order.Proof. Reflexivity is straightforward. Transitivity follows from Lemma 3.2.If we identify all functions which are different up to the permutation, thenrelation ≺ s is antisymmetric, by Corollary 3.2. I isan infinite set We reformulate the notion of linear preservers of submajorization relationon ℓ p ( I ). 14 efinition 4.1. A bounded linear operator T : ℓ p ( I ) → ℓ p ( I ) is called apreserver of submajorization on ℓ p ( I ) + , if T preserves the submajorizationrelation, that is, T f ≺ s T g , whenever f ≺ s g , where f, g ∈ ℓ p ( I ) + . The set ofall linear preservers of submajorization on ℓ p ( I ) + is denoted by P s ( ℓ p ( I ) + ). Theorem 4.1.
Let D ∈ pDsS ( ℓ p ( I )) , p ∈ [1 , ∞ ) , and suppose that Θ := { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } (19) is a family of one-to-one maps on I with disjoint images, where I is atmost a countable set. Then there is at least one S ∈ pDsS ( ℓ p ( I )) such that P θ D = SP θ , ∀ θ ∈ Θ .Proof. Let D ∈ pDsS ( ℓ p ( I )). There are two operators D ∈ DS ( ℓ p ( I )) and D ∈ DsS ( ℓ p ( I )) such that D = D + D (20)by Lemma 3.1. Now, using [6, Lemma 4.2] there exists operator S ∈ DS ( ℓ p ( I )) such that P θ D = S P θ , ∀ θ ∈ Θ. Actually, we can see in theproof of above mentioned theorem that operator S is defined by h S e j , e i i = h D e θ − ( j ) , e θ − ( i ) i , i, j ∈ θ ( I ) for some θ ∈ Θ , , i, j
6∈ ∪ θ ∈ Θ θ ( I ) and j = i, , otherwise.In the similar way, using [20, Theorem 3.2] there is an operator S ∈ DsS ( ℓ p ( I )) such that P θ D = S P θ , ∀ θ ∈ Θ, defined by h S e j , e i i = h D e θ − ( j ) , e θ − ( i ) i , i, j ∈ θ ( I ) , for some θ ∈ Θ ,a, i, j
6∈ ∪ θ ∈ Θ θ ( I ) , and j = i, , otherwise,where 0 ≤ a ≤
1. Clearly, operator S is not uniquely determined. Wedefine bounded linear operator S := S − S . The operator S has form h Se j , e i i = h D e θ − ( j ) , e θ − ( i ) i − h D e θ − ( j ) , e θ − ( i ) i , i, j ∈ θ ( I ) for some θ ∈ Θ , − a, i, j
6∈ ∪ θ ∈ Θ θ ( I ) and j = i, , otherwise. (21)Obviously, S ∈ pDsS ( ℓ p ( I )) by the above representation (21) and the de-composition (20). Now, P θ D = P θ ( D − D ) = P θ D − P θ D = S P θ − S P θ = SP θ . heorem 4.2. Let I be an infinite set and let p ∈ (1 , ∞ ) . Then, P w ( ℓ p ( I ) + ) ⊂P s ( ℓ p ( I ) + ) holds.Proof. Let T ∈ P w ( ℓ p ( I ) + ). Using Theorem 2.4 we get T = X k ∈ I λ k P θ k , where ( λ k ) k ∈ I ∈ ℓ p ( I ) + , I ⊂ I is at most countable and for each k ∈ I , θ k ∈ Θ = { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } .Let f ≺ s g . There is an operator D ∈ pDsS ( ℓ p ( I )) such that f = Dg .Using Theorem 4.1, there is an operator S ∈ pDsS ( ℓ p ( I )) such that P θ D = SP θ , ∀ θ ∈ Θ. Hence,
T f = X k ∈ I λ k P θ k ( f ) = X k ∈ I λ k P θ k ( Dg )= X k ∈ I λ k SP θ k ( g ) = S X k ∈ I λ k P θ k ( g ) = S ( T g ) , thus T f ≺ s T g , so T ∈ P s ( ℓ p ( I ) + ).In the sequel, we will show that every linear preserver of the subma-jorization ( ≺ s ) preserve the weak majorization ( ≺ w ), when p ∈ (1 , ∞ ) andwhen I is an infinite set. Theorem 4.3.
Let T ∈ P s ( ℓ p ( I )) where I is an infinite set and p ∈ (1 , ∞ ) .Then for each pair of distinct elements j , j ∈ I , functions T e j and T e j are different up to the permutation and T e j ( i ) · T e j ( i ) = 0 , for all i ∈ I .Proof. Since e j ≺ s e j and e j ≺ s e j , we have T e j ≺ s T e j and T e j ≺ s T e j . Using Corollary 3.2, functions T e j and T e j are different up to thepermutation.In order to show second part, we will suppose contrary that there aretwo elements j , j ∈ I and there exist k ∈ I such that T e j ( k ) · T e j ( k ) = 0.Let c = T e j ( k ) = 0 and c = T e j ( k ) = 0. Since the ”column” T e j in thematrix form of operator T is in ℓ p ( I ), we have lim k →∞ T e j ( k ) = 0. Hence,there is a finite set defined by C := { i ∈ I : T e j ( i ) = c } . Clearly, C is a non-empty set because k ∈ C .16et j ∈ I \ { j , j } . For arbitrary chosen a > b > ae j + be j ≺ s ae j + be j and ae j + be j ≺ s ae j + be j . Therefore, aT e j + bT e j ≺ s aT e j + bT e j and aT e j + bT e j ≺ s aT e j + bT e j . Now, usingCorollary 3.2, functions aT e j + bT e j and aT e j + bT e j are different up tothe permutation, that is ac + bc = aT e j ( k ) + bT e j ( k ) ∈ { aT e j ( i ) + bT e j ( i ) : i ∈ I } . Since the above set is at most countable, using Lemma 2.1 for n = 2 weobtain that there exist k ∈ I such that T e j ( k ) = c and T e j ( k ) = c , sowe conclude that k ∈ C . Since j is arbitrary chosen element of the infiniteset I \ { j , j } and since the set C is finite, there is at least one element k ∈ C for which there is an infinite set T ⊂ I such that T e j ( k ) = c and T e t ( k ) = c , ∀ t ∈ T . Now, for adjoint operator T ∗ : ℓ q ( I ) → ℓ q ( I ) of T ,where q is conjugate exponent of p , we get k T ∗ e k k q = X j ∈ I | T ∗ e k ( j ) | q ≥ X t ∈T |h T ∗ e k , e t i| q = X t ∈T |h T e t , e k i| q = X t ∈T c q = ∞ , which is impossible.In the other words, we conclude that the ”row” k of operator T containsinfinite nonzero elements which are mutually equal. It implies that the sameholds for appropriate k ”column” of adjoint operator T ∗ . Since T ∗ : ℓ q ( I ) → ℓ q ( I ) have to be T ( e k ) ∈ ℓ q ( I ), which is a contadiction with the above fact.Using above two theorems we obtain that preservers of weak majoriza-tion ( ≺ w ) and submajorization ( ≺ s ) on ℓ p ( I ) + coincide when I in an infiniteset and when p ∈ (1 , ∞ ). Using Theorem 2.4, we get the following charac-terization of submajorization preservers on ℓ p ( I ) + . Corollary 4.1.
Let p ∈ (1 , ∞ ) , and let I be an infinite set. Suppose that T : ℓ p ( I ) → ℓ p ( I ) is a bounded linear operator. The following statementsare equivalent:i) T ∈ P s ( ℓ p ( I ) + ) .ii) T e j ≺ s T e k and T e k ≺ s T e j , ∀ k, j ∈ I , and for each i ∈ I there is atmost one j ∈ I such that h T e j , e i i > . ii) T = P k ∈ I λ k P θ k , where ( λ k ) k ∈ I ∈ ℓ p ( I ) + , I ⊂ I is at most countable, θ k ∈ Θ := { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } . In the sequel, we will find the proper form of linear preservers of subma-jorization on ℓ ( I ), when I is an infinite set. Theorem 4.4.
Let I be an infinite set. Then, P w ( ℓ ( I ) + ) ⊂ P s ( ℓ ( I ) + ) holds.Proof. Let T ∈ P w ( ℓ ( I ) + ). Using Theorem 2.5 statement ii ), we get thatthere is a decomposition of the operator T in the follofing way: T = T + T , where T ∈ P ( ℓ ( I ) + ) and T ∈ P ( ℓ ( I ) + ), where sets I , I ⊂ I aredisjoint with I ∪ I = I and operators T , T are chosen to be h T f, e i i = h T f, e i i = 0 , ∀ i ∈ I , ∀ i ∈ I , ∀ f ∈ ℓ ( I ) + . (22)Using (10), we have T = X k ∈ I λ k P θ k (23)where I is at most a countable subset of I , ( λ k ) k ∈ I ∈ ℓ ( I ) + and for every k ∈ I we have θ k ∈ Θ = { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } . If there is i ∈ I such that λ i = 0 then we will consider the set I \ { i } instead of I . Because of this, we may assume that λ j >
0, for every j ∈ I .Suppose that f ≺ s g for fixed f, g ∈ ℓ ( I ) + . It follows that there exists D ∈ pDsS ( ℓ ( I )) such that f = Dg . Using Theorem 4.1 we conclude thatthere is S ∈ pDsS ( ℓ ( I )) such that P θ k D = SP θ k , ∀ k ∈ I . Obviously, operator S is not unique and it is defined by (21), where 0 ≤ a ≤
1. Similarly as in Theorem 4.2 we obtain that T preserve submajorizationrelation: T f = T Dg = X k ∈ I λ k P θ k ( Dg ) = X k ∈ I λ k SP θ k ( g ) = S ( T g ) . (24)18ext, changing the order of summation we obtain k f k = X i ∈ I | f ( i ) | = X i ∈ I f ( i ) = X i ∈ I X j ∈ I h De j , e i i g ( j )= X j ∈ I X i ∈ I h De j , e i i g ( j ) = X j ∈ I g ( j ) X i ∈ I h De j , e i i ≤ k g k . Thus, inequality k f k ≤ k g k holds. If we set a := 1 − k f kk g k , using the aboveargument we get that 0 ≤ a ≤ . Now, using (11) we get T Dg = T f = h X i ∈ I f ( i ) = h k f k = h (1 − a ) k g k = (1 − a ) T g, (25)where h := T e j , for some j ∈ I .We claim that ST = (1 − a ) T . (26)Firstly, we will show that Se k = (1 − a ) e k , for every k ∈ I . Fix k ∈ I . Wehave that h T f, e k i = 0, by (22). Since, h T f, e k i = X j ∈ I f ( j ) h T e j , e k i = X j ∈ I f ( j ) X i ∈ I λ i h P θ i e j , e k i = X j ∈ I f ( j ) X i ∈ I λ i h e θ i ( j ) , e k i It follows that k
6∈ ∪ i ∈ I θ i ( I ), because f is arbitrary fixed and λ is positive.Now, using the definition (21) of the operator S we get Se k = (1 − a ) e k .Now, using (22) we obtain ST u = ( X j ∈ I u j ) Sh = ( X j ∈ I u j ) X k ∈ I h ( k ) Se k = (1 − a )( X j ∈ I u j ) X k ∈ I h ( k ) e k = (1 − a ) T u, for every u ∈ ℓ ( I ) + , so (26) is provided. Finally, using (24), (25) and (26)we obtain T f = ( T + T ) Dg = T Dg + T Dg = ST g +(1 − a ) T g = ST g + ST g = ST g.
Since, S ∈ pDsS ( ℓ ( I )), we get T f ≺ s T g , that is T ∈ P s ( ℓ ( I ) + ).19n order to show that P s ( ℓ ( I ) + ) is a subset of P w ( ℓ ( I ) + ) we need thefollowing lemmas. Lemma 4.1.
Let T ∈ P s ( ℓ ( I ) + ) . Suppose that Q is a finite subset of I and let ∆ : Q → Q be a bijection. For every a ∈ I there is b ∈ I such that h T e i , e a i = h T e ∆( i ) , e b i , ∀ i ∈ Q. (27) Proof.
Since, X i ∈ Q a i e i ≺ s X i ∈ Q a i e ∆( i ) and X i ∈ Q a i e ∆( i ) ≺ s X i ∈ Q a i e i we get X i ∈ Q a i T e i ≺ s X i ∈ Q a i T e ∆( i ) and X i ∈ Q a i T e ∆( i ) ≺ s X i ∈ Q a i T e i for each ( a i , a i , . . . , a i m ) with a i j > i j , where m = card ( Q ) ∈ N .We get that functions P i ∈ Q a i T e i and P i ∈ Q a i T e ∆( i ) are different up to thepermutation by Corollary 3.2, that is X i ∈ Q a i h T e i , e a i ∈ X i ∈ Q a i h T e ∆( i ) , e k i | k ∈ I . Since codomain of the positive operator T is ℓ ( I ) we have T e j ∈ ℓ ( I ) + andcard(Im( T e j )) ≤ ℵ , so P i ∈ Q a i T e ∆( i ) is at most a countable set. We getthat for fixed a ∈ I there is a b ∈ I such that (27) holds, by Lemma 2.1. Lemma 4.2.
Let T ∈ P s ( ℓ ( I ) + ) , where I is an infinite set. If there aretwo distinct n, m ∈ I such that h T e n , e r i > and h T e m , e r i > for some r ∈ I , then h T e n , e r i = h T e m , e r i .Proof. Suppose that there exist m, n, r ∈ I such that h T e m , e r i > h T e n , e r i > h T e m , e r i 6 = h T e n , e r i .Let a , a > l ∈ I \ { m, n } be arbitrary chosen. Clearly, a e m + a e n ≺ s a e m + a e l and a e m + a e l ≺ s a e m + a e n . Because T ∈ P s ( ℓ ( I ) + ) we obtain a T e m + a T e n ≺ s a T e m + a T e l and a T e m + a T e l ≺ s T a e m + a T e n .
20t follows that a h T e m , e r i + a h T e n , e r i ∈ { a h T e m , e j i + a h T e l , e j i | j ∈ I } . by Corollary 3.2.Since a T e m + a T e l ∈ ℓ ( I ) + , the above set is at most countable, sousing Lemma 2.1 for n = 2 we get that for l there is k ∈ I such that h T e m , e k i = h T e m , e r i and h T e l , e k i = h T e n , e r i . (28)On the other hand, it is clear that for fixed c ∈ R , c = 0 holds card { i ∈ I | h T e m , e i i = c } < ℵ . (29)Since I is an infinite set and l ∈ I \ { m, n } is arbitrary chosen, using (28)and (29) there is s ∈ I and there is a sequence ( t i ) i ∈ N of distinct elements t i ∈ I such that h T e m , e s i = h T e m , e r i and h T e t i , e s i = h T e n , e r i , ∀ i ∈ N . Let { Φ j } j ∈ N be a family where Φ j := { t , t , . . . t j } for every j ∈ N .We define bijections γ j : { t , t , . . . t j } ∪ { m } → { t , t , . . . t j } ∪ { m } inthe following way: γ j ( x ) := t j , x = m,m, x = t j x, x ∈ { t , t , . . . t j − } . For each j ∈ N there exists r j ∈ I such that h T e m , e r j i = h T e γ j ( t j ) , e r j i = h T e t j , e s i = h T e n , e r i , (30) h T e t j , e r j i = h T e γ j ( m ) , e r j i = h T e m , e s i = h T e m , e r i , (31)by Lemma 4.1 Also, for each x ∈ { t , t , . . . t j − } we get h T e x , e r j i = h T e γ j ( x ) , e r j i = h T e x , e s i = h T e n , e r i , (32)again by Lemma 4.1.If we provide that the set { r j | j ∈ N } is countable, we obtain using (30)that h T e m , e r j i = h T e n , e r i , for all j ∈ N which is a contradiction with (29).Let k < k for some integers k , k and suppose that r k = r k . Sincebijections γ k and γ k are different, using (31) for γ k we obtain h T e t k , e r k i = h T e m , e r i . k < k we get using (32) for γ k that h T e t k , e r k i = h T e t k , e r k i = h T e n , e r i . Combine above two facts, we get h T e m , e r i = h T e t k , e r k i = h T e n , e r i which is a contradiction with the assumption at the beginning of the proof h T e m , e r i 6 = h T e n , e r i . Thus, r k = r k for all k , k ∈ N , so the set { r j | j ∈ I } is a countable. Theorem 4.5.
Let I be an infinite set. Then, P s ( ℓ ( I ) + ) ⊂ P w ( ℓ ( I ) + ) holds.Proof. Let T ∈ P s ( ℓ ( I ) + ). Because e i ≺ s e j and e j ≺ s e i it follows that T e i ≺ s T e j and T e j ≺ s T e i . Since, relation ≺ s implies ≺ w , we have T e i ≺ w T e j and T e j ≺ w T e i , so the first part of statement iv ) in Theorem 2.5 issatisfied.Suppose that there exist m, n, r ∈ I such that h T e m , e r i > h T e n , e r i >
0. Using Lemma 4.2 we get that h T e m , e r i = h T e n , e r i . Precisely, all non-zero elements in one ”row” have to be mutually equal.We claim that all elements in the ”row” indexed by r are the same, thatis, there is no zero element. Suppose contrary that there exists l ∈ I suchthat h T e l , e r i = 0. Fix k ∈ I \ { m, n, l } . We define a bijection θ k ( x ) := m, x = m,n, x = n,k, x = l,l, x = k. Now, applying Lemma 4.1 on bijection θ k we get that there exists i k ∈ I such that h T e m , e i k i = h T e m , e r i , h T e n , e i k i = h T e n , e r i and h T e k , e i k i = h T e l , e r i = 0. Using T e m , T e n ∈ ℓ ( I ) + , it follows thatcard { i ∈ I | h T e m , e i i = h T e n , e i i = h T e m , e r i} < ℵ . (33)It is easy to see that i k is contained in the above set. Now, since k isarbitrary chosen from infinite set I \ { m, n, l } , there exists at least one s ∈ I ( s is contained in the set considered in (33)) and there is a sequence ( t i ) i ∈ N
22f distinct elements t i ∈ I such that h T e m , e s i = h T e m , e r i = h T e n , e s i = h T e n , e r i and h T e t i , e s i = 0, ∀ i ∈ N . Now, for each j ∈ N we define bijections γ j : { t , t , . . . t j } ∪ { m, n } → { t , t , . . . t j } ∪ { m, n } , by γ j ( x ) := m, x = m,t j , x = n,n, x = t j ,x, x ∈ { t , t , . . . t j − } . Similarly as in (30), (31) and (32), for each j ∈ I and for the suitablebijection γ j , there exists r j ∈ I such that h T e m , e r j i = h T e m , e s i = h T e m , e r i , (34) h T e t j , e r j i = h T e γ j ( n ) , e r j i = h T e n , e s i = h T e n , e r i , (35) h T e n , e r j i = h T e γ j ( t j ) , e r j i = h T e t j , e s i = 0 , (36)again by Lemma 4.1. Also, for every x ∈ { t , t , . . . t j − } we have h T e x , e r j i = h T e γ j ( x ) , e r j i = h T e x , e s i = 0 . (37)by Lemma 4.1.Suppose that k < k . We will show that r k = r k . Using (37) for γ k ( x ) we get that h T e t k , e r k i = 0 . However, using (35) for bijection γ k ( x ) we obtain h T e t k , e r k i = h T e n , e r i > , so we get that h T e k , e r k i 6 = h T e k , e r k i , therefore r k = r k .Finally, we get that sequence { r j } j ∈ N contains mutually district mem-bers. Using (34) it follows that k T e m k p = X i ∈ I h T e m , e i i ≥ X j ∈ I h T e m , e r j i = ∞ X i =1 h T e m , e r i = + ∞ , which is impossible. Thus, there is no l ∈ I such that h T e l , e r i = 0, so theset {h T e j , e i i | j ∈ I } is a singleton. It follows that operator T satisfiesstatement iv ) of Theorem 2.5, that is T ∈ P w ( ℓ ( I ) + ).23sing Theorem 4.4 and Theorem 4.5 we obtain that preservers of weakmajorization ( ≺ w ) and submajorization ( ≺ s ) on ℓ ( I ) + coincide when I inan infinite set. Using Theorem 2.5, we get the following characterization ofsubmajorization preservers on ℓ ( I ) + . Corollary 4.2.
Let T : ℓ ( I ) → ℓ ( I ) be a bounded linear operator, where I is an infinite set. The following statements are equivalent:(i) T ∈ P s ( ℓ ( I ) + ) ;(ii) There are operators T ∈ P ( ℓ ( I ) + ) and T ∈ P ( ℓ ( I ) + ) and disjointsets I , I ⊂ I with I ∪ I = I such that T = T + T where T , T arechosen to be h T f, e i i = h T f, e i i = 0 , ∀ i ∈ I , ∀ i ∈ I , ∀ f ∈ ℓ ( I ) + ; (iii) There is an at most a countable set I ⊂ I and there is a family Θ := { θ k : I − −−→ I | k ∈ I , θ i ( I ) ∩ θ j ( I ) = ∅ , i = j } of one-to-one maps, θ k ∈ Θ , ∀ k ∈ I , and ( λ i ) i ∈ I ∈ ℓ ( I ) + such that T = X k ∈ I λ k P θ k + T h , where T h ( f ) := h P k ∈ I f ( k ) , for h ∈ ℓ ( I ) + with h h, e j i = 0 , ∀ j ∈ S i ∈ I θ i ( I ) ;(iv) T e j ≺ w T e k and T e k ≺ w T e j , ∀ k, j ∈ I , and for each i ∈ I , eitherthere exists exactly one j ∈ I with h T e j , e i i > or the set {h T e j , e i i| j ∈ I } is a singleton. We recall that the set of all linear preservers of weak majorization on ℓ p ( I ), p ∈ [1 , ∞ ) is a norm-closed by [20, Theorem 4.4] and [21, Theorem4.2]. Linear preservers of weak majorization and submajorization coinside,by Corollaries 4.1 and 4.2, hence the following corollary is straightforward. Corollary 4.3.
Let I be an infinite set, and let p ∈ [1 , ∞ ) . The set P s ( ℓ p ( I ) + ) is a norm-closed subset of the set of all bounded linear opera-tors on ℓ p ( I ) . xample 4.1. We define maps θ i : N → N by θ i ( j ) = i + 1 + i + j − X k =0 k, ∀ i, j ∈ N . (38)Suppose that there exist i , i , j , j ∈ N such that θ i ( j ) = θ i ( j ) . (39)It follows that i + i + j − X k =0 k = i + i + j − X k =0 k. (40)If i + j < i + j then, by (40), we get i − i = i + j − X k = i + j − k ≥ i + j − . Since j , i ∈ N , we get by above that j + i ≤ i + j > i + j then we get j + i ≤ i + j = i + j . Now, (40) gives i = i and j = j . Therefore θ i ( N ) ∩ θ i ( N ) = ∅ , for all i , i ∈ N with i = i . Also, using definition (38)it is easy to see that maps θ i are one-to-one for each i ∈ N . Thus,Θ := { θ k : N − −−→ N | k ∈ I , θ i ( N ) ∩ θ j ( N ) = ∅ , i = j } . (41)Let T be an operator defined by T = ∞ X i =1 λ i P θ i , (42)where λ = ( λ i ) i ∈ N ∈ ℓ ( N ) + is an arbitrary fixed function. The operator T T = . . .λ . . . λ . . .λ . . . λ . . . λ . . .λ . . . λ . . . λ . . . λ . . .λ . . . λ . . . λ . . . λ . . . λ . . .λ . . . ... ... ... ... ... . . . Using (4) we obtain
T f := [0 , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , . . . ] T for each f ∈ ℓ p ( N ). The operator T is a bounded linear operator on ℓ p ( N ),for all p ∈ [1 , ∞ ), by Theorem 2.1.Using ℓ ( N ) + ⊂ ℓ p ( N ) + , we get that operator T satisfies the statement iii ) of Corollary 4.1. Also, using the matrix representation of T we mayconclude that T satisfies statement iv ) of Corollary 4.2. Thus, T preservesthe submajorization relation on ℓ p ( N ) + , for each p ∈ [1 , ∞ ). Let T ( f ) := h X i ∈ N f ( i ) , ∀ f ∈ ℓ ( N ) (43)where h ∈ ℓ ( N ) + defined by h ( j ) := (cid:26) a ≥ , j = 1 , , otherwise . Now, the operator T := T + T T = a a a a a . . .λ . . . λ . . .λ . . . λ . . . λ . . .λ . . . λ . . . λ . . . λ . . .λ . . . λ . . . λ . . . λ . . . λ . . .λ . . . ... ... ... ... ... . . . and T f := " a ∞ X i =1 f ( i ) , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , λ f , . . . T for each f ∈ ℓ ( N ). The operator T is a bounded linear operator on ℓ ( N )(by [22, Theorem 3.1]) because it satisfies (5). Now, T preserves the sub-majorization relation on ℓ ( N ) + by statement iii ) (or iv )) in Corollary 4.2.In the above example, we presented linear preservers of submajorizationon ℓ ( N ) + which have only one ”row” with mutually equal non-zero elements.In the next example we give preservers where sets I and I in Corollary 4.2are both countable, that is, where there are countable ”rows” which are asingleton. Example 4.2.
Let θ i : N → N be maps defined by θ i ( j ) = i − i + j − X k =1 k, ∀ j ∈ N (44)for any i ∈ N . Similarly as in the previous example, we may provide that thefamily (41) contains one-to-one maps θ i ( j ) with mutually disjoint images.27ix a non-zero sequence µ = ( µ i ) i ∈ N ∈ ℓ ( N ) + and suppose that h ∈ ℓ ( N ) + is defined by h ( j ) := µ i , j = i +1 P k =2 k, , otherwise . In this way, we define the operator T := T + T , where operators T and T are determined as in (42) and (43), respectively.In order to provide that statement iii ) of Corollary 4.2 is valid, supposecontrary that there exist r ∈ ∪ i ∈ N θ i ( N ) such that h h, e r i >
0. It follows that r = θ i ( j ) and r = n +1 P k =2 k for some i, j, n ∈ N . Hence, we get n +1 X k =2 k = r = i − i + j − X k =1 k that is, n +1 X k =1 k − i + j − X k =1 k = i ≥ . Therefore, n + 1 > i + j −
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M. Z. Ljubenovi´c: [email protected]
D. S. Raki´c: [email protected]@gmail.com