Toeplitz operators and univalent functions

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 T_wf = 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 ||T_1g+T_1u || > 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 C_gf = 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 C_g is unitarily equivalent to T_ng, where n_g(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 ||C_g|| ≤ 1.

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.