## Compact operators on sequence and function spaces: Characterisations and duality results

##### Abstract/Overview

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 [6]) 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 [10], [22], [25], [26J and [27]), 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 [8], [19J and [20]).
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.