Abstract
The project will study class of n-power operators and their properties. The study will focus n-power normal operators, n-power hyponormal operators, n- power posinormal operators, n-power quasi-normal operators and n-power quasi- isometry operators. We shall look at their basic properties as well as their spectral and numerical properties.
Table of Contents
Acknowledgement iii
Dedication iv
Abstract v
Chapter One: Introduction
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . 2
Chapter Two
2.1 n-Power normal operators . . . . . . . . . . . . . . . . . . . . . . 6
2.2 n-power hyponormal operators . . . . . . . . . . . . . . . . . . . 14
2.3 n-power posinormal operators . . . . . . . . . . . . . . . . . . . . 20
2.4 n-power Quasi-normal operators . . . . . . . . . . . . . . . . . . 25
2.5 n-power quasi-isometry . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter Three: Spectral properties of n-power operators
3.1 Spectral properties n-power normal operators . . . . . . . . . . . 36
3.2 Spectral properties of n-power hyponormal operators . . . . . . . 37
3.3 Spectral properties of n-power posinormal operators . . . . . . . 37
3.4 Spectral properties of n-power quasi-isometry . . . . . . . . . . . 38
Chapter Four:
Numerical Ranges of N-power operators 40
5 References 43
Chapter One
Introduction
1.1 Literature Review
Operator theory including its subtopics such as spectral theory came into focus after 1900.Around this time, Fredholm’s report on the theory of integral equa- tions was published. He gave a complete analysis of integral equations which he referred to as Fredholm’s equations that extended results from linear algebra to a class of operators. He also defined the determinant to a class of operators and was the first to use the term resolvent operator.
1902, Lebesque introduced an important category of spaces known as LP . Around the same time, Hilbert founded spectral theory as a result of a series of articles by Fredholm. Now, the word ”spectrum” was adopted by Hilbert in 1897 from an article by Wilhelm Wirtinger. Hilbert used the notion of integral equations and found results underL2spaces,square-integrable functions and discussed also some results for the scenario where integral operator was symmetric. In 1906, Hilbert discovered continuous spectrum,a work away from integral equations.
Around 1913, Frigyes Riez,introduced the concept of algebra of operators where he studied of bounded operators on the Hilbert space L2.In his work,he intro- duced other concepts like Riez representation theorem, orthogonal projectors and spectral integrals.
In 1916, Riez found the theory of completely continuous operators now referred to as compact operators. He further extended Fredholm’s work on the spectral theorem of compact operators. Further developments came in between 1929-32 when spectral theorem of self-adjoint and normal operators were discovered by Marshall Stone and John Von Neumann.
Neumann also introduced concepts that are widely used in operator theory like closure of an operator, adjoint operators, unbounded operators and extension of operators. In 1932, Stefan Banach published a first text on operator theory which included the closed-graph theorem, Weak convergence and the fixed point theorem.
Israil Gel’fand in 1941 extended the spectral theorem to elements of a normed algebra and introduced the spectral radius formula as well as C - algebra and the character of an algebra.
Since Gel’fand’s time, operator theory has been an enormous branch of Mathne- matics. Many authors have defined new classes of operators and new interesting results have been captured. Patel and Ramanujan (1981) introduced and stud- ied normal operators. An operator T is called normal if T T = TT
Adnan Jibril(2008) extended the notion of normal operators to n-power nor- mal operators and showed that an operator T B(H) is n-power normal operator if and only if Tn is normal. An operator T is n-power normal if T commutes with Tn, that is TnT = T Tn.
A. brown in 1953 introduced the concept of quasi-normal operator. An op- erator T is quasi-normal if T commutes with T T . Sid Ahmed(2011) studied the concept of n-power quasi-normal operators as an extention quasi-normal oper- ators. An operator TB(H) is n-power quasi-normal if Tn commutes with T T .
Panayappan and Sivaman(2012) coined the concept of n-binormal operators.P.R Halmos and Stampfli introduced hyponormal operators in 1962. T is called hy- ponormal if TT ≥T T
Guesba et al. extended the concept of hyponormal operators to n-power hy- ponormal operators in 2016. An operator T is called n-power-hyponormal if TT ≥T T
.
In this project, we shall study the fundamental properties of n-power operators. For each class of n-power operator, we shall study their subclasses and spectral and numerical range properties where applicable.
1.2 Notation and Definitions
Notations
Hilbert spaces will be denoted by H and B(H) will denote the algebra of bounded operators in the Hilbert space H. T and S will denote operators and I will denote the identity operator on the Hilbert space.
The spectrum, the point spectrum and the residual spectrum of an operator T will be denoted by σ(T ), σa(T ) and σp(T ) respectively.
The residual spectrum of T is denoted by σr(T ), Ker(T) is used as the ker- nel of T and λ as the eigenvalue of T.
The range of T will be denoted R(T ) while the nullity of T will be denoted by N (T )
Definitions
Definition
Let X be a vector space over a field of complex numbers.A norm on X is a mapping ∥.∥ : X → C such that it satisfies the following axioms
1) ∥ax∥ = |c|∥x∥c ∈ C, x ∈ X
2) ∥x∥ ≥ 0
3)∥x∥ = 0 if and only if x = 0 for all x ∈ X
4) x + y x + y for all x, y X
Definition
Let T inB(H),then there exists an operator T ∗ known as the adjoint of T such that < x, Ty >>=< T ∗x, y > for all x, y ∈ H
An operator T∈ B(H) is said to be
Self-adjoint if T = T ∗ Normal if T ∗T = TT ∗ Isometry if T ∗T = I
Quasinormal if T (T ∗T ) = (T ∗T )T Hyponormal if T ∗T ≥ TT ∗
Subnormal if it has a normal extension Binormal if T ∗T and TT ∗
commute Unitary if T ∗T = TT ∗ = I
Partial isometry if T = TT ∗T
Seminormal if either T or T is hyponormal Involution if T 2 = I
Idempotent if T 2 = T
Co-isometry if TT ∗ = I
Nilpotent if Tn = 0 for some n
Quasi-nilpotent if σ(T ) = {0}
Definition
An operator T is called a normaloid if and only if the spectral radius is equal to its operator norm. i.e r(T ) = ∥T ∥
Definition
An operator T is called posinormal if it has a positive interrupter or in other words if there exists a positive operator P such that the self commutator [T ∗, T ] of [T ∗, T ] = T ∗(I − P )T
Definition
An operator Lk is called an integral operator with the kernel k is defined by
Lkf (x) = k(x, y)f (y)dy
Definition
An operator T is called a quasi-isometry if T ∗2T 2 = T ∗T
Definition
An operator P is called an interrupter for T if TT ∗ = T ∗PT
Definition
The commutator of two operators K and L which is denoted by [K, L] can de- fined as [K, L] = KL - LK
Definition
Two bounded linear operators K and L in the Hilbert space H are said to be unitarily equivalent if there exists a unitary operator A G(K, L) such that AK = LA
Definition
The set of all λ C such that λI T is not invertible is called the spectrum of T and λ C is an eigenvalue of the operator T if there exists x H Definition
For λ σ(T ) such that λI T is not bounded from below is known as the point spectrum of T.
Definition
For λ σ(T ) such that λI T is one-to- one but unbounded from below is called the approximate spectrum of T.
Definition
For λ σ(T ) such that λI T has no dense range is known as the residual spectrum of T.
Furthermore, An operator T ∈ B(H) is called: n-power normal operator if TnT ∗ = T ∗Tn. n- power hyponormal if T ∗Tn ≥ TnT ∗.
n-power Quasinormal if TnT ∗T = T ∗TTn. n-power posinormal if R(Tn) ⊂ R(T ).
n-power quasi-isometry if Tn−1T ∗2T 2 = T ∗TTn−1 for some integer n.
Definition
T B(H) is called an isoloid if every isolated point of the σ(T) is a member of the point spectrum of T.
Definition
λ ∈ C is an eigenvalue of T if ker(TλI) 0
Definition
The Numerical range W (T ) of an operator T is a subset of the complex num- ber C given by: W (T ) = {< Tx, x >, x ∈ H, ∥x∥ = 1 with the property that W (αI + βT ) = α + βW (T )for all α, β ∈ C
Definition
The numerical radius denoted by r(T ) of an operator T on the Hilbert space H is given by r(T ) = sup{|λ| : λ ∈ W (T )} i.e sup {| < Tx, x > |, ∥x∥ = 1}
Definition
A set X is called convex if for every two points x, y ∈ X we have tx+(1−t)y ∈ X
for all t ∈ [0, 1].
Definition
The convex hull of an operator T is the smallest convex set containing
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