Na primeira parte desta tese nós estudamos os operadores de convolução Tµ que são tauberianos agindo nas álgebras de grupo L1(G), onde G é um grupo abeliano localmente compacto e µ é uma medida de Borel complexa sobre G. Nós mostramos que esses operadores são invertíveis se o grupo G não é compacto e que eles são de Fredholm quando têm imagem fechada e G é compacto. Além disso, se G é compacto nós provamos que Tµ é de Fredholm se a parte singular contínua de µ respeito à medida de Haar de G é zero. Na segunda parte nós estudamos os operadores de convolução Tµ que são cotauberianos em L1(G). Nós mostramos que esses operadores são tauberianos e são de Fredholm (de índice zero). Além disso, mostramos que Tµ é tauberiano se, e somente se, sua extensão natural à álgebra de medidas M(G) é tauberiano. Mostramos alguns resultados obtidos por dualidade de espaços de Banach para os operadores de convolução tauberianos e cotauberianos agindo sobre C0(G), o espaço de Banach das funções complexas que se anulam no infinito, e L&#8734(G), o espaço de Banach das funções mensuráveis essencialmente limitadas. Finalmente estendemos alguns dos resultados obtidos para álgebras de Banach que possuem uma identidade aproximada limitada. / In the first part of this thesis we study the convolution operators Tµwhich are tauberian as operators acting on the group algebras L1(G), where G is a locally compact abelian group and µ is a complex Borel measure on G. We show that these operators are invertible when G is non-compact, and that they are Fredholm when they have closed range and G is compact. Moreover, if G is compact, we prove that Tµ is Fredholm when the singular continuous part of µ with respect to the Haar measure on G is zero. In the second part we study the convolution operators Tµ which are cotauberian as operators acting on L1(G). We show that these operators are tauberian and Fredholm of index zero. Moreover, we show that Tµ is tauberian as an operator on L1(G) if and only if so is its natural extension to the algebra of measures M(G). We show some results, obtained by duality, about tauberian and cotauberian convolution operators on the Banach spaces L&#8734(G) of equivalence classes of essentially bounded mesurable functions on Gand C0(G) of complex valued continuous functions on Gwhich vanish at infinity. Finally, we extend some results obtained to Banach algebras with a bounded identity approximate.

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For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.

We describe a new algebra of boundary value problems which contains Lopatinskii elliptic as well as Toeplitz type conditions. These latter are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. Every elliptic operator is proved to admit up to a stabilisation elliptic conditions of such a kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. The crucial novelty consists of the new type of weighted Sobolev spaces which serve as domains of pseudodifferential operators and which fit well to the nature of operators.

Let X, Y be locally compact Hausdorff spaces. A linear operator T from C0(X,E) to C0(Y,F) is called disjointness preserving if coz(Tf)¡äcoz(Tg) = whenever coz(f)¡äcoz(g) = ∅. We discuss some cases on these compact disjointness preserving operators T and prove that if £f0 is a nonzero point of £m(T), then £f0 is an eigenvalue of T and we find a projection ∏: C0(X,E) ¡÷C0(X,E), such that for Y1 = ∏C0(X;E) and Y2 = (1-∏)C0(X;E), the operator T|Y1 -£f0 is a nilpotent and £f0-T|Y2 is invertible.

Let X be a compact Hausdorff topological space. The Banach space C(X) consists of all continuous complex value functions with the supnorm. An operator P on C(X) is called a generalized bicircular projection if P + £f(I − P) is an isometry for all |£f| = 1, £f in C and P2 = P. In this thesis, we study some projections which are the averages of two composition operators or two weighted composition operators on C(X). If a projection is the average of the identity and a composition operator, it is a generalized bicircular projection. And give an example of a projection which is the average of the identity and a weighted composition operator, but not a generalized bicircular projection. We also discuss some projections which are the average of two bounded linear operators on a Banach space. And the main result is that, let T1 and T2 are two bounded linear operators on a Banach space, and Q = T1+T2 2 . If T1 ¡CT2 = T2 ¡CT1 and T2 1 = T2 2 = Id then Q is a tripotent, i.e. Q3 = Q.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.

The theory of semigroups of linear operators forms an integral part of Functional Analysis with substantial applications to many fields of the natural sciences. In this study we are concerned with the application to equations of mathematical physics. The theory of semigroups of bounded linear operators is closely related to the solvability of evolution equations in Banach spaces that model time dependent processes in nature. Well-posed evolution problems give rise to a semigroup of bounded linear operators. However, in many important and interesting cases the problem is ill-posed making it inaccessible to the classical semigroup theory. One way of dealing with this problem is to generalize the theory of semigroups. In this thesis we give an outline of the theory of two such generalizations, namely, C-regularized semigroups and B-bounded semigroups, with the inter-relations between them and show a number of applications to ill-posed problems. / Thesis (Ph.D.)-University of Natal, Durban, 2001.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. 137-138).