Range-Kernel orthogonality and elementary operators on certain Banach spaces


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Introduction
Let ( ) B be the algebra of all bounded linear operators acting on a complex separable Hilbert space and be a compact operator, and let ( ) ( ) ≥ ≥…≥ s X s X 0 1 2 denote the eigenvalues of | | ( ) = * X X X 1 2 arranged in their decreasing order. (1) where tr denotes the trace functional. Hence, ( ) C 1 is the trace class, ( ) C 2 is the Hilbert-Schmidt class and the case = ∞ p corresponds to the class of compact operators ( ) ∞ C equipped with the norm∥ ∥ ( ) = ∞ X s X 1 . For more details, the reader is referred to [10].
We recall the definition of Birkhoff-James's orthogonality in Banach spaces [2,12]. Definition 1. If is a complex Banach space, then for any elements ∈ x y , , we say that x is orthogonal to y, noted by ⊥ x y, iff for all ∈ α β , there holds If M and N are linear subspaces in X, we say that M is orthogonal to N , noted by ⊥ M N , if∥ ∥ ∥ ∥ + ≥ x y x for all ∈ x M and all ∈ y N . If { } = M x , we simply write ⊥ x N . (i) The orthogonality in this sense is not symmetric; (ii) If is a Hilbert space with its inner product < > . , then it follows from (2) that < > = x y , 0 which means that Birkhoff-James's orthogonality generalizes the usual sense in Hilbert space.
We also recall the definition of the Range-Kernel orthogonality for a pair of operators ( ) E T , on Banach spaces introduced by R. Harte [11].
and → T Y Z : are bounded linear operators between Banach spaces. T is called orthogonal to E provided b i i n are n-tuples of algebra elements. The length of E is defined to be the smallest number of multiplication terms required for any representation ∑ a xb j j j for E. The case of the elementary operator is restricted to the operator , which is well-known as generalized derivation induced by fixed operators A, B in ( ) B and defined by Anderson [5] proved that if A and B are normal operators, then where δ ker A B , denotes the kernel of δ A B , . This means that the kernel of δ A B , is orthogonal to its range. This result has been generalized in different directions, to nonnormal operators (see [6,7] and to some elementary operators (see [8,9]). The main purpose of this paper is to give necessary conditions and characterize the elements that are orthogonal to the range of an operator defined on an abstract reflexive and strictly convex Banach space. As an application, we consider certain classes of elementary operators defined on the spaces ( ) . In addition to the notations and the definitions already introduced, we set, if is a normed linear space over a field = or , we denote by ( ) B the space of all linear bounded operators on , the closure of the range of an operator ( ) ∈ T B will be denoted by ( ) T ran , the restriction of T to an invariant subspace M will be denoted by | T M and the commutator − AB BA of the operators A B , will be denoted by [ ] A B , .

Main results
Let be a normed linear space over the field and † its topological dual. Let D be the (multivalued) mapping defined from to † as: called the normalized duality mapping. Hahn-Banach's theorem ensures that there always exists at least one support functional (a support functional ψ at ∈ x is a norm-one linear functional in † such that ( ) ∥ ∥ = ψ x x at each vector ∈ x ) and therefore ( ) D x is non-empty for every ∈ x . Moreover, it is well known that ( ) D x is convex and weak*-compact subset of † . D is not linear in general but it is homogeneous, : .
Theorem 3. Let K be a closed linear subset of and ∉ x K , then ker .
Proof. Let x y , for all ∈ y K . Hence, ⊥ x K . For the converse, let ∉ x K such that ⊥ x K . Then, for all ∈ y K , x and y are linearly independent vectors. Let L be the closed subspace spanned by K and { } x , , and define the function φ on L by ( ) ∥ ∥ + = φ αx βy α x 2 for all ∈ y K and all ∈ α β , . Clearly φ is linear (by the assumption that K is a linear subset of ). To prove the continuity of φ, let ∈ z L, then = + z αx βy and ( ) ∥ ∥ = φ z α x 2 . By the definition of ⊥ and the assumption that : k e r .
Notation. Let K be a nonempty subset of and ( ) ∈ T B , we denote the duality adjoint of T by T † and set

Remark 5. It is clear that if { }
x n n is a sequence in a subset K converging to y and ⊥ x x n , for all n, then ⊥ x y. Hence, ⊥ ⇒ ⊥ x K x K.
Lemma 6. Let be a Banach space.
1. If K and L are closed subspaces of and ⊆ ⊥ K L r , then ⊕ K L is closed.
Proof. Then, { } x n n is a Cauchy sequence, hence x lim n n exists in K . Setting = x x lim n n , we get = − ∈ y z x L lim n n , and therefore, ∈ ⊕ z K L. 2. (i) By Theorem 3 and Remark 5,

Let
It is a direct consequence of assertions (1) and (2)(i). □ be a map on , not necessarily linear or additive, and → + F : f be a map defined by We say that F f has a global minimum at ∈ a if As an application of the previous results, the following theorem gives us necessary and sufficient conditions in terms of Birkhoff-James orthogonality for minimizing the map F f .
be bounded maps not necessarily linear or additive, and ∈ a . Then the following assertions hold: Since the maps f T , satisfy the relation cited in (i), the sufficient condition follows from (i). By linearity of T , we get ( ) ( ) ( ) The other equivalence follows from Lemma 1. If ( ) f a is a smooth point (or the space is smooth), then ( ) f a has only one functional support (for the functional support of smooth points, see [10]) and therefore ( ( )) D f a has one element (the map D is a single-valued function if is smooth). □

Lemma 8. [8]
If is a separable ideal of compact operators in ( ) B equipped with unitary invariant norm, then its dual † is isometrically isomorphic to an ideal of compact operators not necessarily separable as follows: and f F , f are defined as in Theorem 7(ii) and where ( ) f A is given by its polar decomposition ( ) Then, the following assertions hold , then F f has a global minimizer at A if and only if (

Proof. From Theorem 7, we have that F f has a global minimizer at A if and only if there exists
, then by the isomorphism in (8), it follows that

Range-Kernel orthogonality results  275
(ii) It is well known that ( ) C 1 is not reflexive, even not smooth space and its dual ( ) . This isomorphism is given by Then, and tr for all .
So that if ( ) f A is a smooth point, then F f has a global minimizer at A iff there is a unique operator R such that Since ( ) f A is smooth, then by Holub [11], either ( ) f A or ( ) * f A is injective, thus either u or * u is an isometry, i.e., = * uu I or = * u u I . So it suffices to take Let J be the natural injection between and † † , i.e., ; , : ; . † † † , if T † is orthogonal. Then Proof. Let ∈ s such that ⊥ s T ran . Then, by Lemma 6(2.i) and the smoothness of , there is a unique ( ) ∈ φ D s s such that ∈ φ T ker s † . Again, by assumptions and Lemma 6(2.i), there is Let J be the natural injection between and † † as defined in (10). We see that ( ) ∥ ∥ = J s φ φ s s 2 and ∥ ( )∥ ∥ ∥ = J s φ s , which means that ( ) ( ) ∈ J s D φ s . By the reflexivity of , J is a bijection. Hence, there is If is a reflexive separable Banach space and T † is orthogonal, then the implication (11) holds with respect to suitable norm in . Indeed, if is separable, then there is an equivalent norm, which is smooth and strictly convex in .
Corollary 13. Let be a reflexive, smooth and strictly convex Banach space and ( ) ∈ T B . If T and T † are orthogonal, then Proof. If T is orthogonal, then, by Definition 2, it follows that s  s  T  s  T  : ker ran and the reverse implication of (12) follows by Proposition 1. Let us prove the decomposition (13): let ∈ y such that ( ( ) ) ∈ ⊕ ⊥ y T T ker ran r , then there is , for all ∈ s T ker and all ∈ x . For = s 0, it follows, by Lemma 6(2.i), that ⊥ Y T ran , and by Proposition 1, ⊆ T T ran ker . So that, we can choose = x 0 and = s y, this yields ( ) = φ y 0 y . This means ⊥ y y, and hence = y 0. Finally, the decomposition (13) follows from Lemma 10. □

Applications
In this section, we consider the important case, when the operator T , cited in the previous section, is replaced by the elementary operators are n-tuples in ( ( )) B n . By the isomorphisms (8) and (9), we can assert that the duality adjoint of E on ( ) ( ) Lemma 14. Let , be Hilbert spaces, such that ( ) = E X AXB. If A and * B are injective operators, then E is injective. Proof.
(ii) See [12]. □ As an application of the previous section, we shall give certain necessary conditions and characterization of the operators in ( )