Spectral Decomposition of Compact Operators on Hilbert Spaces.
ABSTRACT
This research work focuses on the study of Compact linear operators and Spectral decompositions on Hilbert spaces. This is broken into three chapters where
In chapter one, the properties and characterization of bounded linear operators on finite and infinite dimensional vector space will be studied.
The notion of complete normed vector space which is called a Banach space will be studied where we shall show that this notion of completeness is a metric space concept. Examples to illustrate this ideas will be given.
All results results discussed in this chapter are preliminary and can be found in any material on the subject. The material for this chapter follows from Kreyszaig (2007).
In chapter two, we shall study the concept of Compact linear operators on Banach spaces as a map T ∈ L(X, Y ) such that for any bounded subset A ⊂ X, T (A) is compact in Y. Other characterization of this map T will also be discussed.
Fredholm Alternative for Compact linear operators will be discussed and we shall also give examples of Compact Operators such as – The injection of H1(Ω) into L2(Ω).
The Hilbert-Schmidt Operator T : L2(Ω) −→ L2(Ω) with K ∈ L2(Ω × Ω) defined by (Tf )x = Ω K(x, y)f (y)dy for a.e x ∈ Ω, etc. Almost all the results from this chapter are from Brezis (2010), Kreyszig (2007), Rudin (1976) and C.E Chidume (2014).
In chapter three, spectral properties and decomposition of Compact linear operators, classifica- tion of λ ∈ σ(T ) (σ(T) is spectrum of T ) and examples,compact self-adjoint operators and their spectrum will also be discussed, where we obtain eigenvalues and eigenfunctions of compact linear operator from analytic and from their weak formulations.
The materials for the chapter follows from Khalil (2010),Khalil (2017) and Joel (2011).
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TABLE OF CONTENTS
Certi cation i
1 Linear Operators and Boundedness 3
1.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Examples of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Examples of linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Bounded linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Examples of bounded operators on in nite dimensional spaces . . . . . . . . . . . . 10
1.6 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Some properties of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.1 Examples of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Compact linear Operators on Banach spaces 18
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Spectral Decomposition of Compact operators on Hilbert spaces 28
3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Classi cation of 2 (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5.1 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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INTRODUCTION
Compact operators are linear operators on Banach spaces that maps bounded set to relatively compact sets. In the case of Hilbert space H it is an extension of the concept of matrix acting on a finite dimensional vector space.
In Hilbert space, compact operators are the closure of the finite rank operators in the topology induced by the operator norm.
In general, operators on infinite dimensional spaces feature properties that do not appear in the finite dimension case; i.e matrices.
The compact operators are notable in that they share as much similarity with matrices as one can expect from a general operator.
Spectral decomposition of compact operators on Banach spaces takes the form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, the spectral properties of compact operators resembles those of square matrices.
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BIBLIOGRAPHY
Brezis, H. (2010). Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer. C.E, Chidume (2014). Applicable Functional Analysis. African University of Science and Technol-ogy, Abuja., first edition.
Joel, F. (2011). Spectral theory Examples. First edition.
Khalil, E. (2010). Lecture Notes on Functional Analysis and Evolution Equations. African Uni- versity of Science and Technology, Abuja.
Khalil, E. (2017). Lecture Notes on Sobolev spaces and distributions. African University of Science and Technology, Abuja.
Kreyszig, E. (2007). Introductory Functional Analysis with Applications. Wiley classics library. Wiley India Pvt. Limited.
Rudin, W. (1976). Principles of mathematical analysis. McGraw-Hill Book Co., New York, third edition. International Series in Pure and Applied Mathematics.