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<sub>j=0</sub><sup>n</sup r<sub>j</sub>
where each r<sub>j</sub> is a rectifiable Jordan curve and r<sub>i</sub> ∩ r<sub>j</sub> 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|<sub>∂V</sub> with respect to harmonic measure for L is
ν|<sub>∂V</sub> = μ<sub>a</sub>of⁻¹|<sub>∂V</sub> + μ<sub>s</sub>of⁻¹|<sub>∂V</sub>
where V = intL and μ = μ<sub>a</sub> + μ<sub>s</sub> is the Lebesgue decomposition of μ with respect to m.
Applying Sarason’s process, we obtain P<sup>∞</sup>(ν) ≠ L<sup>∞</sup>(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|<sub>Γ</sub> << μ<sub>a</sub>of⁻¹|<sub>Γ</sub>. 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|<sub>Γ</sub> << μ<sub>a</sub>of⁻¹|<sub>Γ</sub>.
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<sup>∞</sup>(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<sup>*n</sup>→0 sot. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/80999 |
| Date | January 1982 |
| Creators | Miller, Thomas L. |
| Contributors | Mathematics, Olin, Robert F., Thomson, J.E., Fletcher, Peter, Farkas, Daniel R., Arnold, Jesse T. |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iii, 76, [2] leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 9185319 |
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