No abstract

Abstract. We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.

We use the concept of angular derivative and the hyperbolic metric in the unit disk ‫,ބ‬ to study the dynamical aspects of the equilibrium points belonging to ‫ބ∂‬ of some complex-analytic dynamical systems on ‫.ބ‬ Our results show a deep connection between the dynamical properties of those equilibrium points and the geometry of certain simply connected domains of ‫.ރ‬ As a consequence, and in the context of semigroups of analytic functions, we give some geometric insight to a well-known inequality of Cowen and Pommerenke about the angular derivative of an analytic function.

We characterize the boundedness and compactness of weighted composition operators between weighted Banach spaces of analytic functions //° and H£°. We estimate the essential norm of a weighted composition operator and compute it for those Banach spaces W° which are isomorphic to CQ. We also show that, when such an operator is not compact, it is an isomorphism on a subspace isomorphic to c 0 or looFinally, we apply these results to study composition operators between Bloch type spaces and little Bloch type spaces.2000 Mathematics subject classification: primary 47B38; secondary 30D45,46E15.

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [5], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [5]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.