The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented
here treats different areas of functional analysis such as spaces of holomorphic
functions, infinite dimensional holomorphy and dynamics of operators.
After a first chapter that introduces the notation, definitions and the basic results
we will use throughout the thesis, the text is divided into two parts. A first one,
consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire
functions on Banach spaces, and a second one, corresponding to Chapters 3 and
4, where we consider differentiation and integration operators acting on different
classes of weighted spaces of entire functions to study its dynamical behaviour. In
what follows, we give a brief description of the different chapters:
In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach
space X, we consider the weighted inductive limits of spaces of entire functions
VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally
in the study of growth conditions of holomorphic functions and have been investigated
by many authors since the work of Williams in 1967, Rubel and Shields
in 1970 and Shields and Williams in 1971. We determine conditions on the family
of weights to ensure that the corresponding weighted space is an algebra or
has polynomial Schauder decompositions. We study Hörmander algebras of entire
functions defined on a Banach space and we give a description of them in terms of
sequence spaces. We also focus on algebra homomorphisms between these spaces
and obtain a Banach-Stone type theorem for a particular decreasing family of
weights. Finally, we study the spectra of these weighted algebras, endowing them
with an analytic structure, and we prove that each function f ¿ VH(X) extends
naturally to an analytic function defined on the spectrum. Given an algebra homomorphism,
we also investigate how the mapping induced between the spectra
acts on the corresponding analytic structures and we show how in this setting
composition operators have a different behavior from that for holomorphic functions
of bounded type. This research is related to recent work by Carando, García,
Maestre and Sevilla-Peris. The results included in this chapter are published by
Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space
of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to
find a predual and to prove that VH(X) is regular and complete. We also study
conditions to ensure that the equality VH0(X) = VH(X) holds. At this point,
we will see some differences between the finite and the infinite dimensional cases.
Finally, we give conditions which ensure that a function f defined in a subset
A of X, with values in another Banach space E, and admitting certain weak
extensions in a space of holomorphic functions can be holomorphically extended
in the corresponding space of vector-valued functions. Most of the results obtained
have been published by the author in [13].
The rest of the thesis is devoted to study the dynamical behaviour of the following
three operators on weighted spaces of entire functions: the differentiation operator
Df(z) = f (z), the integration operator Jf(z) = z
0 f(¿)d¿ and the Hardy
operator Hf(z) = 1
z z
0 f(¿)d¿, z ¿ C.
In Chapter 3 we focus on the dynamics of these operators on a wide class of
weighted Banach spaces of entire functions defined by means of integrals and
supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿,
and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman
spaces of entire functions, denoted by Hv(C) and H0
v (C) if, in addition, p = ¿.
We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic
or uniformly mean ergodic; thus complementing also work by Bonet and Ricker
about mean ergodic multiplication operators. Moreover, for weights satisfying
some conditions, we estimate the norm of the operators and study their spectrum.
Special emphasis is made on exponential weights. The content of this chapter is
published in [17] and [15].
For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿
Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and
chaos. The chapter ends with an example provided by A. Peris of a hypercyclic
and uniformly mean ergodic operator. To our knowledge, this is the first example
of an operator with these two properties. We thank him for giving us permission
to include it in our thesis.
The last chapter is devoted to the study of the dynamics of the differentiation and
the integration operators on weighted inductive and projective limits of spaces of
entire functions. We give sufficient conditions so that D and J are continuous on
these spaces and we characterize when the differentiation operator is hypercyclic,
topologically mixing or chaotic on projective limits. Finally, the dynamics of these
operators is investigated in the Hörmander algebras Ap(C) and A0
p(C). The results
concerning this topic are included by Bonet, Fernández and the author in [16].
Beltran Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral no publicada]. Universitat Politècnica de València. doi:10.4995/Thesis/10251/36578.
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