For any analytic self-map ϕ of {z : |z| < 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cϕ to be closed-range on the Bloch space B. Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cϕ is closed-range on the Bergman space A 2 , then it is closed-range on B, but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem. Mathematics Subject Classification (2010). Primary 47B33, 47B38; Secondary 30D55.Let D denote the unit disk {z : |z| < 1} and let T denote the unit circle {z : |z| = 1}. We let A denote two-dimensional Lebesgue measure on D. The Bergman space A 2 is the collection of functions f that are analytic in D such thatAs a closed subspace of L 2 (A), A 2 forms a Hilbert space with respect to the inner product := D f gdA. The Bloch space B is the collection of functions f that are analytic in D such thatNow ||·|| B defines a norm on B, and under this norm B forms a Banach space. Moreover, ||f || A 2 ≤ 3||f || B for any function f that is analytic in D, and hence B ⊆ A 2 . A function ϕ that is analytic in D and that satisfies ϕ(D) ⊆ D is

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In 1998, John B. Conway and Liming Yang wrote a paper [11] in which they posed a number of open questions regarding the shift on P t (µ) spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review some of the solutions, mention some alternate approaches and discuss further the problem that remains unsolved.2010 Mathematics Subject Classification. Primary 47A15; Secondary 30C85, 31A15, 46E15, 47B38.

We show that for any ε > 0 and any region G whose outer boundary equals {z : |z| = 1}, there is a sequence { n } ∞ n=1 of pairwise disjoint closed disks in G such that {z : |z| = 1} is the set of accumulation points of { n } ∞ n=1 , ∞ n=1 radius( n ) < ε and ω (harmonic measure on the boundary of := {z : |z| < 1} \ (∪ ∞ n=1 n ) for evaluation at some z o in ) is supported on ∪ ∞ n=1 (∂ n ).

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