الفهرس 
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Abstract
The complex analysis has fascinated mathematicians since Cauchy,
Weierstrass and Riemann had built up the field from their different
points of view. One of the essential problems in any area of
mathematics is to determine the distinct variants of any object
under certain considerations. The theory of function spaces plays
an important role not only in complex analysis but also in the most
branches of pure and applied mathematics, e.g. in approximation
theory, partial differential equations, geometry and mathematical
physics (see [4, 10]). Quaternion analysis is one of the possible
generalizations of the theory of analytic functions in one complex
variable to Euclidean space (see [1–3, 11]).
Operator theory on different spaces of analytic functions have
been actively appearing in different areas of mathematical sciences
like dynamical systems, theory of semigroups, isometries and quantum
mechanics (see [12]).
The present thesis deals with the theory of complex and hypercomplex
function spaces. First, we apply some certain operators to
analytic and hyperbolic function spaces. Moreover, for the product
operators Jg,SÁ(I g,SÁ) and ThCÁ, necessary and sufficient conditions
are given to be bounded operators in some complex spaces. Second,
we give characterizations for B¤
®, log and F¤(p, q, s) by the help of
1
PREFACE
Hadamard gap class. Third, we introduce a new class of hyperholomorphic
functions, so-called F®
!,G(p, q, s) spaces. For this class, we
obtain the relations between F®
!,G(p, q, s) spaces and quaternion
B®
! space. characterizations of the hyperholomorphic F®
!,G(p, q, s)
functions by the coefficients of certain lacunary series expansions
in quaternion analysis are given. Finally, we study the existence
of the solutions for linear differential equation in the QK,!(p, q)
spaces. This work is divided into six chapters organized as follows.
In Chapter 1, we provide a suitable groundwork to the types
of complex function spaces which will be needed in this thesis. This
chapter is arranged as follows. In Section 1.1, we begin with background
materials in the complex plane. Section 1.2 deals with some
notations, definitions and results of various classes of analytic and
hyperbolic functions. These classes have been studied in the theory
of Banach function spaces such as B®
log, B®
!, F(p, q, s), QK,!(p, q),
B¤
®, Q¤
K,!, F¤(p, q, s), and Q¤
p spaces. In Section 1.3, we list several
operators like composition operators, Integral operators and product
operators. In Section 1.4, we recall some definitions and results
in Hadamard gaps. Section 1.5 is devoted to quaternion analysis
and their properties. Finally, in Section 1.6, we provide the basic
concepts and results of the theory of complex linear differential
equations in the unit disc.
In Chapter 2, we prove the boundedness of composition operators
acting between some analytic and hyperbolic weighted family
of function spaces. Also, we study the boundedness property of
product extended Cesáro operators and composition operators from
the Bloch-type spaces B® to QK(p, q) spaces on the unit disc D.
2
In Sections 2.1 and 2.2, we study the composition operator CÁ from
Bloch-type B® spaces to QK,!(p, q) spaces and from B¤
® spaces to
Q¤
K,!(p, q) spaces. The criteria for these operators to be bounded
and Lipschitz continuous are given. In Section 2.3, we prove the
boundedness and compactness properties of product of extended
Cesáro operators and composition operators from the Bloch-type
spaces B® to QK(p, q) spaces on the unit disc D.
In Chapter 3, we obtain some characterizations for functions
in B¤
®,log spaces and F¤(p, q, s) spaces in terms of Hadamard gap
class in the unit disc D. In Section 3.1, we obtain characterizations
of the weighted hyperbolic ®- Bloch B¤
®,log functions by the coefficients
of certain lacunary series expansions in the unit disc. In
Section 3.2, characterization of the hyperbolic functions F¤(p, q, s)
using integral representation of Hadamard gaps are given.
In Chapter 4, we study the boundedness of operators Jg,SÁ
and I g,SÁ from the weighted Bergman spaces to the Bloch type
spaces. Also, we prove the boundedness of these operators from
the Bloch-type spaces to the weighted Bergman spaces. In Section
4.1, we introduce a definitions for new product operators Jg,SÁ and
I g,SÁ. In Section 4.2, we characterize the boundedness of the operator
Jg,SÁ :A
p
®
!B¯(B¯
0 ). In Section 4.3, we give the conditions
to prove the compactness of the operator Jg,SÁ : A
p
®
! B¯(B¯
0 ).
In Section 4.4, we characterize the boundedness of the operator
I g,SÁ :A
p
®
!B¯(B¯
0 ). In Section 4.5, we give the conditions to prove
the compactness of the operator I g,SÁ : A
p
®
!B¯(B¯
0 ). In Section
4.6, we characterize the boundedness and compactness of the operator
Jg,SÁ, I g,SÁ :B¯(B¯
0 )!A
p
® .
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PREFACE
In Chapter 5, we define a new class of hyperholomorphic functions
so-called F®
!,G(p, q, s) spaces. We obtain the relations between
F®
!,G(p, q, s) spaces and the quaternion B®
! spaces. Moreover, we
characterize the hyperholomorphic F®
!,G(p, q, s) functions by the
coefficients of certain lacunary series expansions in quaternion
analysis. In Section 5.1, the relations between F®
!,G(p, q, s) and
B®
! spaces are given in quaternion sense. In Section 5.2, we obtain
a sufficient and necessary condition for any hyperholomorphic
function f on the unit ball B1(0) of R3 with Hadamard gaps in
F®
!,G(p, q, s) spaces.
In Chapter 6, we study the existence of the solutions for complex
linear differential equation of the form
f (n)(z)Å An¡1(z) f (n¡1)(z)Å...Å A1(z) f 0(z)Å A0(z) f (z) Æ 0,
with coefficients of analytic functions An in the unit disc. Our
contribution is to build a relationship between the coefficients and
the solutions of this equation. Moreover, sufficient conditions for
the analytic coefficients An such that all solutions in QK,!(p, q)
spaces are given. In Section 6.1, we give results and definitions
to characterize QK,!(p, q) spaces. In Section 6.2, we build up a
relationship between the coefficients and the solutions of the above
equation such that the all solutions in QK,!(p, q) spaces.