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On Orbits of Operators on Hilbert Space
In this dissertation we treat some problems about possible density of orbits for
non-hypercyclic operators and we enlarge the class of known non-orbit-transitive operators.
One of the questions related to hypercyclic operators that we answer is
whether the density (in the set of positive real numbers) of the norms of the elements
in the orbit for each nonzero vector in the Hilbert space is sufficient to imply that
at least one vector has orbit dense in the Hilbert space. We show that the density
of the norms is not a sufficient condition to imply hypercyclicity by constructing a
weighted bilateral shift that, on one hand, satisfies the orbit-density property (in the
sense defined above), but, on the other hand, fails to be hypercyclic. The second
major topic that we study refers to classes of operators that are not hypertransitive
(or orbit-transitive) and is related to the invariant subspace problem on Hilbert space.
It was shown by Jung, Ko and Pearcy in 2005 that every compact perturbation of
a normal operator is not hypertransitive. We extend this result, after introducing
the related notion of weak hypertransitivity, by giving a sufficient condition for an
operator to belong to the class of non-weakly-hypertransitive operators. Next, we
study certain 2-normal operators and their compact perturbations. In particular, we
consider operators with a slow growth rate for the essential norms of their powers.
Using a new idea, of accumulation of growth for each given power on a set of different
orthonormal vectors, we establish that the studied operators are not hypertransitive
Operators on Banach Spaces of Bourgain-Delbaen Type
We begin by giving a detailed exposition of the original Bourgain-Delbaen
construction and the generalised construction due to Argyros and Haydon. We
show how these two constructions are related, and as a corollary, are able to
prove that there exists some and an uncountable set of isometries
on the original Bourgain-Delbaen spaces which are pairwise distance
apart.
We subsequently extend these ideas to obtain our main results. We construct
new Banach spaces of Bourgain-Delbaen type, all of which have dual.
The first class of spaces are HI and possess few, but not very few operators.
We thus have a negative solution to the Argyros-Haydon question. We remark that
all these spaces have finite dimensional Calkin algebra, and we investigate the
corollaries of this result. We also construct a space with Calkin
algebra and show that whilst this space is still of Bourgain-Delbaen type with
dual, it behaves somewhat differently to the first class of spaces.
Finally, we briefly consider shift-invariant preduals, and hint at
how one might use the Bourgain-Delbaen construction to produce new, exotic
examples.Comment: Oxford University DPhil Thesi
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