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Toeplitz Operators and Univalent Functions

Voas, Charles Howard
Thesis/Dissertation; Online
Voas, Charles Howard
Pitt, Loren
Rosenblum, Marvin
Let A² denote the Hilbert space of functions analytic and square integrable in the open unit disk D. For w ∈ L(D), the Toeplitz operator with symbol w is defined for f ∈ A² by Twf = P(wf), where P is the orthogonal projection from L²(D) onto A². A norm estimate is given when w is a sum of indicator functions of domains in D. Specifically, we call a domain G ⊂ D inaccessible if meas(D\G) > 0 and there exists a point z ∈ G and K ≥ 0 so that |p(z)|² ≤ K∬D\G |p|² dxdy for all polynomials p. Otherwise, G is called accessible. This notion is closely related to mean approximation by polynomials and has been studied by several authors. We show that if G is an accessible domain and U is an open subset of G, then ||T1g+T1u || > 1 This result is used to prove a generalization of the Loewner-FitzGerald theorem on boundary values of schlict mappings. Let Ʋ denote the Hilbert space of functions f(z) defined in D which vanish at the origin and have square integrable derivative. For g ∈ Ʋ with ||g|| ≤ 1, the composition operator with symbol g is defined by Cgf = f∘g for f ∈ Ʋ. We use theorems of Hastings and McDonald-Sundberg to characterize the functions g which induce bounded or compact composition operators on Ʋ. The main technique is the observation that C*g Cg is unitarily equivalent to Tng, where ng(w) is the number of solutions of g (z) = w in D. Our generalization of the Loewner-FitzGerald theorem is: If g ∈ Ʋ and ||g|| ≤ 1, and g(D) is accessible, then g is univalent if and only if ||Cg|| ≤ 1.
University of Virginia, Department of Mathematics, PHD, 1980
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Digitization of this thesis was made possible by a generous grant from the Jefferson Trust, 2015. Thesis originally deposited on 2016-03-14 in version 1.28 of Libra. This thesis was migrated to Libra2 on 2017-03-23 16:35:38.
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