Composition Operators Between Harmonic Bloch-Type Spaces

For a weight mu, that is, a positive continuous function defined on the open unit disk D in C, the weighted harmonic Bloch-type space B-H(mu) is the collection of the complex-valued harmonic mappings f defined on D such that ||f||(BH mu) := |f(0)| + sup(z is an element of D)mu(z)(|f(z)(z)| + f(z)(z)|) < infinity, where f(z) and f(z) denote the first complex partial derivatives with respect to z and z. In this work, given an analytic self-map, phi, of D, and a weight mu, we characterize the bounded and the compact composition operators C-phi from a class of Banach spaces X of harmonic mappings on D into the weighted harmonic Bloch-type space B-H(mu). We study in detail the composition operator between the harmonic Bloch-type spaces BH alpha and BH beta whose corresponding weights are (1 - |z|(2))(alpha) and (1 - |z|(2))(beta), where alpha, beta > 0. In particular, we give an approximation of the operator norm and obtain a precise formula when the symbol fixes the origin. We extend some results on the isometries among such operators that are valid on the corresponding subspaces of analytic functions. Furthermore, we provide a formula of the essential norm in terms of the norm of the monomials z(k) and their corresponding images phi(k) under the operator.