Toeplitz operators and univalent functions
Voas, Charles Howard, Department of Mathematics, University of Virginia
Rosenblum, Marvin, Department of Mathematics, University of Virginia
Pitt, Loren, Department of Mathematics, University of Virginia
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.
PHD (Doctor of Philosophy)
Toeplitz operators, Matrices
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.
All rights reserved (no additional license for public reuse)