# Well-bounded operators

Lim, Boon Hee (1974) Well-bounded operators. MSc(R) thesis, University of Glasgow.

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## Abstract

This thesis is primarily concerned with the structure theory of well-bounded operators and the relationships between various classes of well-bounded operators and prespectral operators. In chapter I, we follow Ringrose (11) to discuss well-bounded operators a non-reflexive Banach space, X. It turns out that well-boundedness of T∈L(X) is equivalent to the existence of a family of projections {E(t): t∈R} on X*, called the decomposition of the identity for T, satisfying certain natural properties and such that = b - a dt (x?X, x*∈ x*). In this case, the family {E(t): t∈R} is not necessarily unique, and a necessary and sufficient condition for its uniqueness is given. In chapter II and III, we discuss three subclasses of well-bounded operators. These are well-bounded operators decomposable in X and well-bounded operators of type (A) and type (B). The main results are that if T is a well-bounded operator decomposable in X then it is uniquely decomposable and that if T is a well-bounded operator of type (A), the algebra homomorphism from AC(J) into L(X) can be extended to an algebra homomorphism from NEV(J) into L(X). We also give some examples in the last section. In chapter III, we follow Spain (14) to use an elementary integration theory to establish directly the characterisation of the type (B) operators. (Theorem III.4.3). We also show that if T is a well-bounded operator of type (B) and {F*(t): t∈R} is the unique decomposition of the identity for T, then for f∈AC(J), we have f(T) = ∫a-bf(t) dF(t) where the integral exists as a strong limit of Riemann sums. Moreover, F(s) - F(s-) is a projection on X onto {x : Tx = sx} and the residual spectrum of T is empty. In the fourth and final chapter, we prove some results concerning relationship between various classes of well-bounded operators and prespectral operations. The main, results are that an adjoint of an operator T∈L(X) with 6(T) CR is a scalar-type operator of class X if and only if T is well-bounded with a decomposition of the identity of bounded variation and that a well-bounded spectral operator is a scalar-type spectral operator and of type (B). Moreover, a well-bounded prespectral operator which is decomposable in X is a scalar-type operator. Finally, we give a counter-example showing that there is a well-bounded operator of type (B) which is not a scalar-type spectral operator.

Item Type: Thesis (MSc(R)) Masters Adviser: H R Dowson Mathematics 1974 Enlighten Team glathesis:1974-74147 Copyright of this thesis is held by the author. 23 Sep 2019 15:33 23 Sep 2019 15:33 https://theses.gla.ac.uk/id/eprint/74147