Approximate spectral synthesis in the Bergman space

Abstract

algebra of bounded holomorphic functions in the unit disk) can be approximated by finite Blaschke products, in the topology of uniform convergence on compact subsets of the disk. Along the way, we find a theorem about kernel functions for weighted Bergman spaces of the type that estimate them away from the diagonal. The motivation forthese results is the study of z-invariant subspaces in the Bergman space. It is known that the lattice of z-invariant subspaces in Bergman spaces has a very complicated structure. But one may single out among all z-invariant subspaces those of simplest nature, the zero-based ones. What z-invariant subspaces can be approximated by zero-based ones? What z ∗ -invariant subspaces can be approximated by finite-dimensional ones? In this paper, we answer these questions (see Theorems 1 and 2), and we discuss some relations with rational approximation and cyclic vectors forthe backward shift. Let X be a Banach space of functions analytic in the unit disk D ={ z ∈ C :| z| < 1} of the complex plane C. Suppose that X is invariant with respect to the operator Mz of multiplication by the independent variable. Many important problems concerning the structure of the lattice of subspaces of X that are closed and invariant with respect to Mz (or, simply, z-invariant) are related to problems of spectral synthesis. Let Y be the space dual to X. Then the functionals kλ of evaluation of functions in X at the points of D, kλ : f → f (λ); λ ∈ D, are eigenvectors of the operator M ∗ : � λI − M ∗ z � kλ = 0;