Compact operators on sequence and function spaces: Characterisations and duality results
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Since the early seventies numerous papers appeared in which the authors considered the question "For which Banach spaces X and Y is the space K(X, Y) of compact linear operators uncomplemented in the space L(X, Y) of bounded linear operators?". The most general answer to this question has close connection with the question whether the scalar sequence space Co of null sequences embeds isomorphically into K(X, Y). In one of the early papers ([28J (1979)) J. Johnson followed a path through dual spaces of spaces of operators when he proved that there is a projection on L(X, Y)* with range isomorphic to K(X, Y)* and kernel the annihilator of K(X, Y) when Y has the bounded approximation property. Johnson then applied his result to consider L(X, Y) as an isomorphic subspace of K(X, Y)** and could then derive necessary and sufficient conditions for K(X, Y) to be reflexive, provided that either X or Y has the bounded approximation property. It turns out that the spaces X and Y have to be reflexive and that the property K(X, Y) = L(X, Y) necessarily has to hold for K(X, Y) to be reflexive in this case. The questions described in the previous paragraph led to research into different directions. Much work went into the study of K(X, Y) as a subspace of L(X, Y) and the question when (i.e. for which Banach spaces) is the equality K(X, Y) = L(X, Y) true? For instance an extensive investigation on when K(X, Y) is an M-ideal in L(X, Y) was done. And some researchers (for example in the papers [2J and ) considered the question on the equality of K(X, Y) and L(X, Y) in the context of scalar sequence spaces and Banach function spaces - i.e. where either X or both X and Yare such spaces. Also, especially in recent papers (for example in , , , [26J and ), the same questions and the question about projections from L(X, Y)* onto K(X, Y)* were considered in the setting of Banach spaces which fail the approximation property. These studies also extended to similar research activities in the setting of locally convex spaces (for example in the papers , [19J and ). The objective in the present thesis is to contribute to the above mentioned study, in the following ways: (a) In line with recent developments we want to show the existence of a suitable projection onto the space K(X. Y)*, which will allow us to find necessary and sufficient conditions for the reflexivity of K(X, Y) without relying on the presence of the bounded approximation property on X or Y. The idea is to put recent work of others in connection with continuous dual spaces of spaces of bounded linear operators in a suitable framework and to improve the present known results and techniques. (b) Use techniques from the theory of vector sequence spaces to simplify proofs of existing results in connection with the equality K(X, Y) = L(X, Y) when X is a Banach scalar sequence space and then to extend the existing results to include more general sequence spaces X. (c) Exposing that recent studies in connection with "absolutely summing multipliers" and " sequences in the range of a vector measure" are intertwined, we intend to extend the concept of "absolutely summing multiplier" to more general types of "summing multipliers" and to apply our work to consider properties of Banach space valued operators on scalar sequence spaces. What are our contributions in this thesis? * Firstly, we introduce an operator ideal approach which seems to provide a natural setting in which to consider the existence of projections from L(X, Y)* onto K(X, Y)* and derive necessary and sufficient conditions for the reflexivity of K(X, Y) in the absence of the approximation property. Thus we simplify the proofs of existing results in the literature and also generalise these results to such an extend that at least the well known spaces without the approximation property are included. * Secondly, in line with modern trend to provide proofs for theorems about operators on Banach spaces which do not rely on the approximation properties, we expand the concept of conjugate ideal to introduce the operator dual space of spaces of bounded linear operators. It turns out that the operator dual space is a handy tool to study inclusion theorems for spaces of operators. Also, using operator dual techniques, we are able to prove existing characterisations of continuous dual spaces of important classes of operators without relying on the continuity of the trace functional with respect to the nuclear norm - thus the proofs do not depend on the approximation property. * Thirdly, we provide a direct and easy proof of a known result which provides necessary and sufficient condition for all weak p-summable sequences in a Banach space to be norm null. Our proof uses sequence space arguments only, thereby allowing us to extend the proof to more general sequence spaces, including certain Orlicz sequence spaces. Applying these results, together with some known characterisations of operators on sequence spaces in terms of vector sequence spaces, we succeed on the one hand to provide easier proofs for existing results about necessary and sufficient conditions for the equalities K(1!.P, X) = L(1!.P,X) and K(co, X) = L(co, X) and on the other hand to obtain further improvements. * An absolutely summing multiplier of a Banach space X is a scalar sequence (ai) such that (aixi) is absolutely summable in X for all weakly absolutely summable sequences (Xi) in X. Recently there were several papers by Spanish mathematicians about sequences in the range of a vector measure. We expose the fact that these concepts are intertwined and thereby show that various results in one of the papers about sequences in the range of a vector measure can easily be obtained, using the absolutely summing multiplier concept. In the last chapter of the thesis we generalise the idea of absolutely summing multiplier to that of p-summing multiplier, A- summing multiplier and even more general, (A, E)- summing multiplier and use these concepts to obtain results about Banach space valued bounded linear operators on A.