Universal composition operators on weighted Dirichlet spaces

Abstract

It is known that the invariant subspace problem for Hilbert spaces is equivalent to the statement that all minimal non-trivial invariant subspaces for a universal operator are one dimensional. In this paper, we first give a characterization of the boundedness of composition operators on weighted Dirichlet spaces \({\mathcal {D}}_{\alpha }(\Pi ^{+})\) over the upper half-plane \(\Pi ^{+}\) using generalized Nevanlinna counting functions, where \(\alpha >-1.\) As an application, we discuss the boundedness of composition operators on \({\mathcal {D}}_{\alpha }(\Pi ^{+})\) induced by linear fractional self-maps of \(\Pi ^{+}.\) Second, we characterize composition operators and their adjoints induced by affine self-maps of \(\Pi ^{+}\) that have universal translates on \({\mathcal {D}}_{\alpha }(\Pi ^{+}).\) Moreover, we investigate which composition operators and their adjoints induced by hyperbolic non-automorphism self-maps of the open unit disk \({\mathbb {D}}\) have universal translates on weighted Dirichlet spaces \({\mathcal {D}}_{\alpha }({\mathbb {D}})\) for \(\alpha >-1.\) Finally, we consider the minimal invariant subspaces of the composition operators that have universal translates.