In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.

We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.

AMS 2000 Mathematics subject classification: Primary 47A58; 45C05. Secondary 47B34

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.