2026/08/02 by Paul Bourdon, Mahsa Fatehi
Mathematics · #math.FA #msc:47A53 #msc:47B91 #msc:47A15 #msc:47B01
27 pages, no figures
arxiv created 2026/08/02 · arxiv updated 2026/08/04
We identify properties of a Banach space B of analytic functions on the open unit disk \mathbbD in the complex plane ensuring that a multiplication operator Mψ: B \toB is Fredholm if and only if its symbol ψ is bounded away from 0 near ∂ \mathbbD. The properties we identify are shared by a wide variety of much-studied spaces, including the Hardy spaces Hp(\mathbbD), weighted Bergman spaces Apω(\mathbbD), Hardy-Sobolev spaces H2β(\mathbbD), the spaces Sjp(\mathbbD) of functions having j-th derivative in Hp(\mathbbD), and the disk algebra A. Thus, as a corollary, our work characterizes Fredholm multiplication operators on these spaces. In addition, we describe the closed, finite-codimensional subspaces of B that are invariant under Mz: B → B in terms of the zeros that the functions in such subspaces have in common. We discuss connections between these subspaces and the problem of characterizing the Fredholm multiplication operators on B, and we prove that Mz restricted to such a subspace is always cyclic, with a polynomial cyclic vector having degree equal to the codimension of the subspace.