Multipliers on Spaces of Holomorphic Functions

Abstract

AbstractWe consider multipliers on the space of holomorphic functions of one variable $$H(\Omega ),\,\Omega \subset \mathbb {C}$$ H ( Ω ) , Ω ⊂ C open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to $$\Omega $$ Ω and $$\Omega $$ Ω is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on $$H(\Omega )$$ H ( Ω ) . We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when $$\Omega $$ Ω is convex.