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

If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then ∂U = U_j=0<sup>n</sup r_j where each r_j is a rectifiable Jordan curve and r_i ∩ r_j is a finite set whenever i ≠ j. Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|_∂V with respect to harmonic measure for L is ν|_∂V = μ_aof⁻¹|_∂V + μ_sof⁻¹|_∂V where V = intL and μ = μ_a + μ_s is the Lebesgue decomposition of μ with respect to m. Applying Sarason’s process, we obtain P^∞(ν) ≠ L^∞(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|_Γ << μ_aof⁻¹|_Γ. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|_Γ << μ_aof⁻¹|_Γ. Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H^∞(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S^*n→0 sot. / Ph. D.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

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.

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.

In this report we survey some recent developments of relationship between Hausdorff moment sequences and subnormality of an unilateral weighted shift operator. Although discrete convolution of two Haudorff moment sequences may not be a Hausdorff moment sequence, but Hausdorff convolution of two moment sequences is always a moment sequence. Observing from the Berg and Dur´an result that the multiplication operator on Is subnormal, we discuss further work on the subnormality of the multiplication operator on a reproducing kernel Hilbert space, whose kernel is a point-wise product of two diagonal positive kernels. The relationship between infinitely divisible matrices and moment sequence is discussed and some open problems are listed.