A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators

Abstract

A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cϕ induced by a hyperbolic automorphism ϕ of the unit disc D acting on the classical Hardy space H 2 is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe's Theorem based on analytic Toeplitz operators.

Universal composition operators

“…We end this article with a result that highlights a connection between cyclicity and universality. We note that the best known examples of universal operators are adjoints of analytic Toeplitz operators T ϕ f = ϕf for ϕ ∈ H ∞ , f ∈ H 2 or those that are similar to them (see [8], [9]), and all such coanalytic Toeplitz operators are cyclic (see [19]). Theorem 5.1.…”

“…The classical proof of the automorphic case (see [14]) and that of the recent nonautomorphic case (see [4]) are both elaborate and involved. An elegant proof of the automorphic case was found recently by Cowen and Gallardo-Gutiérrez [8]. Theorem 5.1 suggests a possible approach to simplify both proofs: this approach is based on showing that the range of C ϕ − λ is closed for some eigenvalue λ, rather than proving surjectivity.…”

On Universal Operators and Universal Pairs

“…In our final example, we use the recent approach of Cowen and Gallardo-Gutiérrez [8] to the universality result by Nordgren, Rosenthal and Wintrobe in order to analyse more carefully a pair which shows subtler obstructions related to the existence of Nordgren-Rosenthal-Wintrobe pair.…”

“…In this case, the infinite-dimensionality of Ker(C ϕ − λI) is verified by explicit computation, but the original argument for the surjectivity relies on fairly sophisticated properties of multiplication operators induced by certain Blaschke products in H 2 (D). Only very recently, an alternative argument for the universality of C ϕ − λI on H 2 (D) was given [8]. For other concrete examples of universal operators, see [5,7,25,26].…”

“…Recall from [4, Theorem 5] (see also [8]) that there is an analytic covering map ψ r : D → A r , such that C ϕr − λI and T * ψr−λ are similar operators on By standard duality, the dual version of part (ii) of condition (M) for the pair (T * ψr−λ , T * ψs−μ ) is the requirement that…”