Title of Thesis

Pseudoresolvents and Semigroups

Author(s)

ABDUL SAMI AWAN

Abstract
The notions of resolvent, pseudoresolvent and a few results along with some remarkable properties are recalled. A new concept, the L1-type pseudoresolvent is introduced.
The aim of this work is firstly to give a characterization theorem for L1-type pseudoresolvents and for the generators of L1-type pseudoresolvents. Moreover, the connection between the L1-type pseudoresolvents and C0-equicontinuous semigroups is pointed out.
Secondly, the main part of this work is devoted to approximation of pseudoresolvents and their generators. If Rn; R : ! L(X); n 1 are generated pseudoresolvents and An; A their generators, then it is investigated under which conditions A is approximated by An and R is approximated by Rn; n 1:
In addition, the conditions under which a sequence of generated pseudoresolvents approximates a pseudoresolvent are given, and in this case the connection between generators is studied.
In the last chapter we have proved a theorem of characterization for exponentially bounded semigroups. To any exponentially bounded semigroup we have associated a projective family of semigroups acting on Banach spaces.

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1.1 Introduction1.2 Pseudoresolvents
1.3 Generated Pseudoresolvents
1.4 L1-Type Pseudoresolvents
1.5 Characterization Theorems for Generators of L1-type Pseudoresolvents
1.6 L1-type Pseudoresolvents and C0-Equicontinuous Semigroups

2.1 Introduction

2.2 Approximant Mapping
2.3 Approximation of Generators
2.4 Equicontiuous Sequence of Pseudoresolvents
2.5 Strong Convergence of Sequences of Generated Pseudoresolvent

3.1 Introduction

3.2 Projective Limits of Operators
3.3 Embedding of a Locally Convex Space into a Projective Limit of Banach Spaces
3.4 A Special Subalgebra of L(X)
3.5 Semigroups of Operators of LP (X)

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