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We consider, for G a simply connected domain and 0 < p < ∞, the Hardy space H p (G) formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F | p over the curves τ ({|z| = r}) be bounded for 0 < r < 1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: H p (G) supports compact composition operators if and only if ∂G has finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache [14], who showed that the H p spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with "compact" replaced by "Riesz". We prove similar results for Bergman spaces, with the Hardy-space condition "∂G has finite Hausdorff 1-measure" replaced by "G has finite area." Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded.

Composition operators are used to study the Qp spaces , which coincide with the Bloch space Β for ρ > 1 and are subspaces of BMOA for 0 < ρ < 1. Bounded composition operators from Bergman and Hardy spaces, and from Β to Qp are characterized by function theoretic properties of their inducing maps. Composition operators into weighted Dirichlet spaces and into little-oh subspaces of Qp, and compactness of these operators are also considered. The criteria for these operators to be bounded or compact are then interpreted geometrically when the symbol is univalent.AMS Classification: Primary 47B38 Secondary 30D55, 46E15 Smith-ZhaoHere dA(z) = dxdy/π is Lebesgue area measure normalized so that A(D) = 1. The subspace QPlo of Qp consists of those functions / such that the integral in the display above tends to 0 as |a| -+ 1. The Qp spaces can be thought of as Möbius invariant versions of weighted Dirichlet spaces, in the same way that the space of analytic functions of bounded mean oscillation, BMOA, is a Möbius invariant version of H 2 . For ρ > -1, the weighted Dirichlet space T>p is the Hilbert space of analytic functions on D satisfyingThe same space of functions results if (1 -|z|) is used in place of log(l/|z|) in the integral above. It is clear that this results in an equivalent norm, since these terms are comparable as |z| ->· 1 and the singularity of log(l/|z|) is integrable. In particular, T>o is the classical (unweighted) Dirichlet space, and it is well known that Τ>ι is the Hardy space H 2 and T>2 is the Bergman space A 2 . A change of variables shows that sup 11/ ο σα\\ 2 Vp = \\f\l 2 Qp.Thus the well known characterization (see [8]) of BMOA as those functions in H 2 whose norms after composition with conformai self-maps of the disk remain uniformly bounded is equivalent to the statement that Q\ = BMOA. Furthermore, it is known that when ρ > 1, Qp = β, the Bloch space of analytic functions satisfying ||/||B = sup(l -|z| 2 )|/'(z)| < oo, zÇD and QPl C QP2 if 0 < pi < P2 < 1; see [4] and [7]. Also, Qlfi = VMOA and for ρ > 1, Qpfi = Bo-Here VMOA is the subspace of BMOA consisting of functions of vanishing mean oscillation, and BQ is the "little Bloch" space of functions / analytic on D for which (1 -|z| 2 )|/'(z)| -> 0 as |z| -> 1. Finally, Qp is a Banach space with norm 11/11 = 1/(0)1 + ||/||q" and Qpfi is a closed subspace. Let ψ be an analytic self-map of D and let ϋφ be the induced composition operator. We are interested in the problem of using function theoretic properties of φ to determine when Οψ : <3P2 -> Qpi is bounded or compact, where 0 < p\ < P2-When p2 > 1 we have QP2 = B, and the problem is to characterize when Cv : Β -> Qp is bounded or compact. In this case, working with the Bloch norm, our methods provide results that are nearly complete. This extends recent work of K. Madigan and A. Matheson [12], who provided criteria equivalent to the compactness of Οφ : Β -¥ Β and Cv : Bo Bo. We also note that it is easy to see that Οφ is always bounded on Β and that Οφ is bounded on Bo if and only if φ € Bo...

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This work explores some of the terrain between functional equations, geometric function theory, and operator theory. It is inspired by the fact that whenever a composition operator or one of its powers is compact on the Hardy space H 2 , then its eigenfunctions cannot grow too quickly on the unit disc. The goal is to show that under certain natural (and necessary) additional conditions there is a converse: slow growth of eigenfunctions implies compactness. We interpret the slow-growth condition in terms of the geometry of the principal eigenfunction of the composition operator (the "Königs function" of the inducing map). We emphasize throughout the importance of this eigenfunction in providing a simple geometric model for the operator's inducing map.

Abstract. We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro's theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro's theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.