Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 110-116). Also available via the Internet.

This thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work. In Chapter 2 we study the problem of when P, the set of analytic polynomials, is dense in the Hardy space H^t(G) or the Bergman space L^t_nG, where G is a bounded domain and t ∈ [1,∞). Characterizations of special domains are also given. In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ω. We say a positive measure τ on G is a Carleson measure if there exists a positive constant c such that for each t ∈ [1, ∞) and each polynomial p we have ⎮⎮p⎮⎮_L¹(τ)≤ c ⎮⎮p⎮⎮ _Lᵗ(ω), We characterize all Carleson measures on a normal domain-definition: a domain G where P is dense in H¹(G). It turns out that P is dense in Hᵗ(G) for all t when G is normal. In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation. In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasi similarity under the assumption that G is a normal domain. In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in L²(σ) when R(K) is a hypo dirichlet algebra. In Chapter 7 we illustrate our results by looking at their implications when G' is a crescent. Several interesting function theory problems are studied. In Chapter 8 we study arc length and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let ω be the harmonic measure of G. We show that if J is a rectifiable curve and E ⊂ ∂G ∩ J is a subset with ω(E) > 0, then E has positive length. / Ph. D.

To solve the linear equation Ax = f, where f is an element of Hilbert space H and A is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate -1 the inverse operator A by an operator V which is a polynomial in A. Using the spectral theory of bounded normal operators the problem is reduced to that of approximating a function of the real variable by polynomials of best uniform approximation. We apply two different techniques of evaluating A-1 so that the operator V is chosen either as a polynomial P (A) when P (A) approximates the n n function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn (A) approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A) satisfy three point recurrence relations, thus the approximate solution vectors P (A)f n and Q (A)f can be evaluated iteratively. We compare the procedures involving n Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite. We also show that the technique can be applied to an operator which is not selfadjoint, but close, in the sense of operator norm, to a selfadjoint operator. The iterative techniques we develop are used to solve linear systems arising from the discretization of Freedholm integral equations of the second kind. Both smooth and weakly singular kernels are considered. We show that earlier work done on the approximation of linear functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse operator and are thus special cases of a general result involving an approximation of arbitrary degree to A -1 . / Thesis (Ph.D.)-University of Natal, 1994.

We prove quasisimilar subdecomposable operators without eigenvalues have equal essential spectra. Therefore, quasisimilar hyponormal operators have equal essential spectra. We obtain some results on the spectral pictures of cyclic hyponormal operators. An algebra homomorphism π from <i>H^∞(G)</i> to <i>L(H)</i> is a unital representation for <i>T</i> if <i>π(1) = I</i> and <i>π(x) = T</i>. It is shown that if the boundary of <i>G</i> has zero area measure, then the unital norm continuous representation for a pure hyponormal operator <i>T</i> is unique and is weak star continuous. It follows that every pure hyponormal contraction is in <i>C.₀</i> Let <i>μ</i> represent a positive, compactly supported Borel measure in the plane, <i>C</i>. For each <i>t</i> in [1, ∞ ), the space <i>P^t(μ)</i> consists of the functions in L^t(μ) that belong to the (norm) closure of the (analytic) polynomials. J. Thomson in [T] has shown that the set of bounded point evaluations, <i>bpe μ</i>, for <i>P^t(μ)</i> is a nonempty simply connected region <i>G</i>. We prove that the measure μ restricted to the boundary of <i>G</i> is absolutely continuous with respect to the harmonic measure on <i>G</i> and the space <i>P²(μ)∩C(spt μ) = A(G),</i> where <i>C(spt μ)</i> denotes the continuous functions on <i>spt μ</i> and <i>A(G)</i> denotes those functions continuous on <i>G &macr;</i> that are analytic on <i>G</i>. We also show that if a function <i>f</i> in <i>P²(μ)</i> is zero a.e. <i>μ</i> in a neighborhood of a point on the boundary, then <i>f</i> has to be the zero function. Using this result, we are able to prove that the essential spectrum of a cyclic, self-dual, subnormal operator is symmetric with respect to the real axis. We obtain a reduction into the structure of a cyclic, irreducible, self-dual, subnormal operator. One may assume, in this inquiry, that the corresponding <i>P²(μ)</i> space has <i>bpe μ = D</i>. Necessary and sufficient conditions for a cyclic, subnormal operator <i>S_μ</i> with <i>bpe μ = D</i> to have a self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable. / Ph. D.