Functions of subnormal operators

Description

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. / Ph. D.

Links & Downloads

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

Tags

LD5655.V856 1982.M544

Subnormal operators

Additional Fields

Identifer	oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/80999
Date	January 1982
Creators	Miller, Thomas L.
Contributors	Mathematics
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