Title page for ETD etd-10242005-124055


Type of Document Dissertation
Author Yang, Liming
URN etd-10242005-124055
Title Subnormal operators, hyponormal operators, and mean polynomial approximation
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Olin, Robert F. Committee Chair
Linnell, P. A. Committee Member
Rossi, J. Committee Member
Thomson, J. Committee Member
Wheeler, R. R. Committee Member
Keywords
  • Subnormal operators.
  • Hyponormal operators.
  • Polynomial operators.
  • Approximation theory.
Date of Defense 1993-06-15
Availability restricted
Abstract
We prove quasisimilar sub decomposable 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 H∞( G) to £(H) is a unital representation for T if π(1) = I and π(X) = T. It is shown that if the boundary of G has zero area measure, then the unital norm continuous representation for a pure hyponormal operator T is unique and is weak star continuous. It follows that every pure hyponormal contraction is in C.o

Let μ represent a positive, compactly supported ⨏ Jrel measure in the plane, C. For each t in [1,∞), the space Pt(μ) consists of the functions in Lt (μ) that belong to the (norm) closure of the (analytic) polynomials. J. Thomson in [T] has shown that the set of bounded point evaluations, bpe μ, for P\μ) is a nonempty simply connected region G. We prove that the measure μ restricted to the boundary of G is absolutely continuous with respect to the harmonic measure on G and the space P2(μ) n C( sptμ) = A( G), where C( sptμ) denotes the continuous functions on sptJ1 and A( G) denotes those functions continuous on G that are analytic on G.

We also show that if a function ∫ in P2(μ) is zero a.e. μ in a neighborhood of a point on the boundary, then f 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 p 2 (μ) space has bpeμ = D. Necessary and sufficient conditions for a cyclic, subnormal operator Sμ with bpeμ = D to have a self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable.

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