AlKhezi, Saleh Battah Mohammad
(1980)
Quasispectral operators.
MSc(R) thesis, University of Glasgow.
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Abstract
The main subject in this thesis is the study of quasispectral operators, their basic properties, their roots and logarithms. In Chapter One, which is not claimed as original, we include the main results on the spectral theory of linear operators and the basic properties of prespectral operators which will be required in other chapters. In Chapter Two, we prove the existence of a complex Banach space Y and a homomorphism E(•) from Sigma[p] into a bounded Boolean algebra of projections on Y such that E(•) is not a spectral measure of any class. In Chapter Three we prove that if S is a scalartype operator and A is an operator that leaves invariant all the maximal spectral subspaces of S, then A commutes with S. In Chapter Four, which is the main part of this thesis, we introduce the concept of quasispectral operator. We prove that a quasispectral operator of class T has a unique resolution of the identity of class T and a unique Jordan decomposition for resolutions of the identity of all classes. We show that every prespectral operator of class T is quasispectral of class r but that there exists a quasispectral operator which is not prespectral of any class, We show that a quasispectral operator is decomposable. Vie prove that if a quasispectral operator has a closed range, then so does its scalar part. Finally in this chapter we consider further decompositions of quasispectral operators. In Chapter Five we obtain analogues for caasispectral operators of results in Chapter 10 of [12] on logarithms and roots of prespectral operators, we give an affirmative answer to the following question. Does there exist a prespectral operator T of class T such that f(T) = A, where A is a given prespectral operator of class T. Finally in Chapter Six, we prove a commutativity theorem for alphascalar operators, where alpha = C(K) and K is a compact subset of the complex plane.
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