Functions of subnormal operators

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.

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 Polytechnic Institute and State University

Year dc:date.issued

1982

Author dc:creator

• Miller, Thomas L.

Chair dc:contributor.committeechair

• Olin, Robert F.

Committee members dc:contributor.committeemember

• Thomson, J.E.
• Fletcher, Peter
• Farkas, Daniel R.
• Arnold, Jesse T.

Statement dc:rights

• In Copyright

Licence dc:rights.uri

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

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10919/80999

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/80999