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Subnormal operators, hyponormal operators, and mean polynomial approximation

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dc.contributor.authorYang, Limingen
dc.contributor.committeechairOlin, Robert F.en
dc.contributor.committeememberLinnell, Peter A.en
dc.contributor.committeememberRossi, John F.en
dc.contributor.committeememberThomson, J.en
dc.contributor.committeememberWheeler, Robert L.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T21:22:02Zen
dc.date.adate2005-10-24en
dc.date.available2014-03-14T21:22:02Zen
dc.date.issued1993en
dc.date.rdate2005-10-24en
dc.date.sdate2005-10-24en
dc.description.abstractWe 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<sup>∞</sup>(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.<sub>0</sub></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<sup>t</sup>(μ)</i> consists of the functions in L<sup>t</sup>(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<sup>t</sup>(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>P<sup>2</sup>(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>P<sup>2</sup>(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>P<sup>2</sup>(μ)</i> space has <i>bpe μ = D</i>. Necessary and sufficient conditions for a cyclic, subnormal operator <i>S<sub>μ</sub></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.en
dc.description.degreePh. D.en
dc.format.extentv, 75 leavesen
dc.format.mediumBTDen
dc.format.mimetypeapplication/pdfen
dc.identifier.otheretd-10242005-124055en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-10242005-124055/en
dc.identifier.urihttp://hdl.handle.net/10919/40103en
dc.language.isoenen
dc.publisherVirginia Techen
dc.relation.haspartLD5655.V856_1993.Y364.pdfen
dc.relation.isformatofOCLC# 28872640en
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.lccLD5655.V856 1993.Y364en
dc.subject.lcshApproximation theoryen
dc.subject.lcshHyponormal operatorsen
dc.subject.lcshPolynomial operatorsen
dc.subject.lcshSubnormal operatorsen
dc.titleSubnormal operators, hyponormal operators, and mean polynomial approximationen
dc.typeDissertationen
dc.type.dcmitypeTexten
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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