Subnormal operators, hyponormal operators, and mean polynomial approximation

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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

1993

Author dc:creator

• Yang, Liming

Chair dc:contributor.committeechair

• Olin, Robert F.

Committee members dc:contributor.committeemember

• Linnell, Peter A.
• Rossi, John F.
• Thomson, J.
• Wheeler, Robert L.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

etd-10242005-124055

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/40103