Nonlinear Analysis: Theory, Methods & Applications
Characterizations of some operator spaces by relative adjoint operators
Introduction
In cite [9], we introduce a representation of elements of the function spaces and where is an infinite set and is a normed space (definitions of these spaces are presented in the next section). This takes a similar role to a basis. Hence, we can study separability of the spaces and linear functionals on them more easily. The representation was given by a family of continuous linear functions from into , or . The family may be unordered and so the representation needs unordered summations which are defined in the prerequisites below. Hence we improve Ferrando and Lüdkovsky’s investigation [5], on with geometric aspects. These spaces and especially the -valued bounded function space have been investigated intensively in recent years. Some important references on the space are [2], [3], [4].
Continuous linear operators on these spaces cannot be studied deeply owing to some deficiencies such as the absence of a basis. The space of such operators between some classical Banach spaces such as and was characterized by the unit vector bases of and . For example, a continuous linear operator from into is equivalent, by the mapping , to a sequence such that is weakly unconditionally Cauchy [8, pp. 167], where is the sequence of associate coordinate functionals to the unit vector basis of and is the adjoint operator of . Is there a similar result for the operators from into or ? In this work we give an affirmative answer to this question by introducing the notion of relative adjoint operators and by using the representation of elements of and given in [9]. In fact, we mainly deal with the characterizations of continuous linear operators from arbitrary Banach spaces to the Banach function spaces or .
Section snippets
Prerequisites
Summability of infinite families in normed spaces has a crucial role in this paper. Let be an infinite set, be a family of vectors in a normed space and let denote the family of all finite subsets of . is directed by the inclusion relation and, for each , we can form the finite sum . If the net converges to some in , then we say that the family is summable, or that the sum exists, and we write in . We mean by the convergence of
Relative adjoint operators
Definition 2 Let , , be arbitrary normed spaces and . Then we define ; the -adjoint operator of from into , by such that for each .
This is reduced to the classical definition of the adjoint operator whenever , the scalar field of both and . The -adjoint operators satisfy the usual attributes of the adjoint operators. For example is linear continuous and . Let us prove this equality. The inequality follows as in the
Applications to characterizations of some operator spaces
Theorem 9 Let , be arbitrary Banach spaces and be or . Then the Banach space is equivalent, by the mapping where ; , to the Banach spaceendowed with the norm Proof For each , define Let and say . Then by Lemma 1 Hence is
Acknowledgement
The author is indebted to the referee for valuable comments and suggestions.
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