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One can give various rigorous definitions to the notion of "functional calculus", but a functional calculus is ultimately just a mathematically meaningful way of talking about an operator f(T), where, T is an operator and f is a function. This thesis is concerned with this concept and with one of its applications, the finding of bounds for powers of operators. It is actually this very application that has prompted the entire investigation presented here. This application is relevant to various fields, such as the numerical analysis of PDE and Markov chains. Chapter I presents various abstract approaches to the notion of "functional calculus" that are given content by three major examples: the Riesz-Dunford functional calculus, the Weyl functional calculus and the functional calculus for sectorial operators. Chapter II investigates various conditions that ensure power boundedness for operators, putting the Tadmor-Ritt condition at its center. The Riesz-Dunford calculus is instrumental for the proofs in this chapter. Chapter III investigates Pascale Vitse's use of Cauchy-Stieltjes integrals and their multipliers for obtaining bounds on powers of operators; the chapter closes with an investigation of partially power bounded operators.

Thesis (Ph. D.)--Lehigh University, 1997. / Includes vita. Includes bibliographical references (leaf 46).

Thesis (M. Phil.)--University of Hong Kong, 1993.

Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 67-69).

Thesis--Eidgenossischen Technischen Hochschule in Zurich. / Lebenslauf. Bibliographical footnotes.

The class of pseudo-differential operators with symbols from Sm (superscript) po̧̧ (subscipt)(Ωx IRⁿ) has been extensively studied.The main assumption which characterises this class of symbols is that a(x,Ȩ) є Sm (superscript)po̧̧ (subscipt)(Ωx IRⁿ) should have a polynomial growth in the Ȩ variable only. The x-variable is controlled on compact subsets of Ω. A polynomial growth in both the x and Ȩ variables on a C°°(lR²ⁿ) function a(x,Ȩ) gives rise to a different class of symbols and a corresponding class of operators. In this work, such symbols and the action of the operators on the functional spaces S(lRⁿ) , S'(lRⁿ) and the Sobolev spaces Qs (superscript) (lRⁿ) (s є lRⁿ) are studied. A study of the calculus (i.e. transposes, adjoints and compositions) and the functional analysis of these operators is done with special attention to L-boundedness and compactness. The class of hypoelliptic pseudo-differential operators in IRⁿ is introduced as a subclass of those considered earlier.These operators possess the property that they allow a pseudo- inverse or parametrix. In conclusion. the spectral theory of these operators is considered. Since a general spectral theory would be beyond the scope of this work, only some special cases of the pseudo-differential operators in IRⁿ are considered. A few applications of this spectral theory are discussed

This Dissertation shows how the theory of C*-algebra of graphs relates to the theory of C*-algebras of sofic shifts. C*-algebras of sofic shifts are generalizations of Cuntz-Krieger algebras [8]. It is shown that if X is a sofic shift, then the C*-algebra of the sofic shift, Oₓ, is isomorphic to the C*-algebra of a directed graph E, C *(E). The graph E is shown to be the well known past set presentation of X constructed in [13]. We focus on the consequences of this result: In particular uniqueness of the generators of Oₓ, pure infiniteness, and ideal structure of the algebra Oₓ. We show the existence of an ideal I ⊂ Oₓ such that when we form the quotient, Oₓ/I, it is isomorphic to C*( F), and F is the left Krieger cover graph of X—a well known, canonical graph one can associate with a sofic shift. The dual cover, the right Krieger cover, can also be related to the structure of Oₓ, and we illustrate this relationship. Chapter 6 shows what happens when we label a directed graph E in a left resolving way. When the graph E and the labeling satisfy certain technical conditions, we can generate a C*-algebra Lₓ ⊂ C*(E), with Lₓ ≅ Oₓ provided that X an irreducible sofic shift. / Graduate

Muhly and Solel developed a notion of Morita equivalence for C*- correspondences, which they used to show that if two C*-correspondences E and F are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak* version of this result by considering (weak) Morita equivalence of W*-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of W*-correspondences E and F implies weak Morita equivalence of their Hardy algebras $H^{\infty}(E)$ and $H^{\infty}(F)$. We give special attention to W*-graph correspondences and show a number of results related to their Morita equivalence. We study how different representations of a W*-algebra give rise to Morita equivalent objects. For example, we show that if (E,A) is a W*-graph correspondence and we have two faithful normal representations $\sigma$ and $\tau$ of A, then the commutants of the induced representions $\sigma ^{\ms{F}(E)}(H^{\infty}(E))$ and $\tau ^{\ms{F}(E)}(H^{\infty}(E))$ are weakly Morita equivalent dual operator algebras. We also develop a categorical approach to Morita equivalence of W*- correspondences. This involves building categories of covariant representations and studying the groups $Aut(\mathbb{D}({(E^{\sigma}})^*)$ and $Aut(H^{\infty}(E))$ (the automorphism groups of the unit ball of intertwiners and the Hardy algebra). In this regard, we advance the work of Muhly and Solel by showing new results about these groups, their matrix representation and their algebraic properties.