• Corpus ID: 203587365

EXTENSION AND DECOMPOSITION OF LINEAR OPERATORS DOMINATED BY CONTINUOUS INCREASING SUBLINEAR OPERATORS

@inproceedings{Olteanu2018EXTENSIONAD,
  title={EXTENSION AND DECOMPOSITION OF LINEAR OPERATORS DOMINATED BY CONTINUOUS INCREASING SUBLINEAR OPERATORS},
  author={Octav Olteanu and Janina Mihaela Mihăilă},
  year={2018}
}

• Octav Olteanu, J. Mihăilă
• Published 2018
• Mathematics

moment problem are stated in [10]. Further applications to the classical moment problem are proved in [6], [11] – [15]. The article [13] contains a polynomial approximation result valid on unbounded subsets, whose proof is using Hahn – Banach theorem. The first purpose of this paper is to prove similar results to those from [4] (respectively from [5]), but for spaces of bounded linear operators. Namely, one assumes that the target space is an order complete (Dedekind complete) normed vector lattice. The domain space is an arbitrary normed vector lattice. Thus, one obtains new statements for theorems 2.1, 3.1 [4] and respectively theorems 1, 2 [5] (see section 2). All the theorems in Section 2 refer to applications of Hahn-Banach type results to the Riesz decomposition property in spaces of linear continuous operators. For similar previous results on this subject, see [3] [5], [16] [19]. It is possible that some of the extension results of section 2 to be partially known. In this case, the contribution of Section 2 of the present paper is to give new simple proofs for such results, based on a general earlier result mentioned above [8], [9]. The second purpose of this work is to point out the possibility of decomposition of each element of an equicontinuous family of linear operators as a difference of positive linear operators, such that the corresponding families of the latter (positive) linear operators to be equicontinuous too (section 3). Uniform evaluation of the norms is studied too. Both Sections 2 and 3 are based on the idea of the existence of a dominating increasing sublinear continuous operator. In the end of section 3, under appropriate assumptions, one proves that the space of all bounded linear operators from into is a Dedekind complete Banach lattice. A characteristic of the present work which is new is that of proving results valid for Similar (but not identical) previous results are proved for the spaces (the space of all linear regular operators from into , that is the space of those operators which can be written as a difference of two linear positive operators) and , the space of all operators which are differences of linear continuous positive operators. Finally, the third purpose of this work is to solve constrained optimization problems in infinite dimensional spaces, related to special Markov moment problems [7] (Section 4). The background of the present work consists in some chapters from [16] – [19]. The rest of the article is organized as follows. Section 2 is devoted to Riesz decomposition property in spaces of bounded linear operators and related results. In Section 3, decomposition of linear bounded operators as differences of positive linear bounded operators is investigated. The equicontinuity of the resulting families of linear positive operators is investigated too. Under additional assumption on the target space one proves that the space of all linear bounded operators is an order complete Banach lattice. Section 4 focuses on a constrained optimization problem in infinite dimensional spaces. Section 5 concludes the paper. Extension and decomposition of linear operators [...] increasing sublinear operators 135 2. On Riesz decomposition property for linear bounded operators We start this section by recalling some known results on the subject. A conjecture posed by A. W. Wickstead [3] found a positive answer (Theorem 3.1 [4]). This theorem is a consequence of the following extension – type result. Theorem 2.1 (see [2], Th. 3.5 and [4], Th. 2.1). Let and be Banach lattices, such that is separable and has the countable interpolation property, and let be a continuous sublinear operator. If is a vector subspace of X and U is a linear continuous operator satisfying for all then there exists a linear extension of to all of also satisfying for all Using Theorem 2.1, in [4] one proves the following main positive answer to the conjecture mentioned above. Theorem 2.2 (see [4], Th. 3.1). Let be two Banach lattices such that is separable and has the countable interpolation property. Then the space of all continuous regular operators has the Riesz decomposition property. For statements and proofs of the above theorems formulated in a more general setting see [5]. We recall our necessary and sufficient condition on the extension of a linear positive extension, which generalizes H. Bauer’s theorem ([19], Theorem V. 5.4; see also [8], [9], [10]). Theorem 2.3 (see [8], Th. 2 and [9], Th. III. 2). Let be a preordered vector space of positive cone an order complete vector lattice, a convex operator, a vector subspace, U a linear operator. The following statements are equivalent: (a) admits a linear positive extension such that (b) for all such that Theorem 2.3 was published firstly in [8], without proof. Its detailed proof and related results can be found in [9] (see [9], p. 978 – 980). In the sequel, we deduce some new results from this theorem. Corollary 2.1. Let be as in Theorem 2.3. Assume that verifies the additional condition: for all such that The following statements are equivalent: (a) admits a linear positive extension such that