Spectral Theory of Linear Operators

Abstract

SummaryThis thesis is concerned with the relationship between spectral decomposition of operators, the functional calculi that operators admit, and Banach space structure.The deep connection between the first two of these concepts has long been known.The thesis is organised as follows. Chapter 1 is an introduction to the concepts, ideas and constructions that will be used through at this thesis. Particularly we consider numerical range and hermitian operators which have a critical role in all this thesis.In Chapter 2 we give a brief overview of some of the theory of (strongly) normal (equivalent) operators. Developing the properties of (strongly) normal (equivalent) operators we will show that the possession of a functional calculus on the spectrum of T is equivalent to T' being scalar type prespectral of class X, thus answering a and if either X does not contain a copy of co, or if U and V are decomposable in X, then the representation is unique. We also explore some properties of AC-operators by applying the theory of (Foia §) decomposable operators.Since 1954 the problem of giving sufficient conditions for the sum and product of two commuting spectral operators to be spectral has attracted attention. The boundedness of the Boolean algebra of projections generated by the two resolutions of the identity is critical. then the Boolean algebra of projections generated by £ and F is also bounded.As a consequence of this he showed that the sum and product of two commuting spectral operators is also spectral in each of the above cases. We will show that the weakly closed algebra generated by the real and imaginary parts of a finite family of commuting scalar-type spectral operators on a Banach lattice not containing co, and on a closed linear subspace of a p-concave Banach lattice, where p< oo, is a W*-algebra, and that every operator in this algebra is a scalar-type spectral operator.

On M-hypercyclic semigroup

“…In all that follows X will be a infinite-dimensional separable complex Banach space and B(X) denotes the Banach algebra of all bounded linear operators on X. let T ∈ B(X), T is said to be hypercyclic if there is a vector x ∈ X whose orbit under T , Orb(T, x) = {T n x : n ≥ 0} is dense in X, in this case x is called a hypercyclic vector. For more details about the orbits of operators see [13] and [18]. The first example of a hypercyclic operator was shown by Birkhoff for the translation operator, Rolewicz constructed the first example of operators on Banach space considering B the backward shift on l p , he showed that λB is hypercyclic if and only if |λ| > 1 (see [3]).…”