Transfer of similarity to Hilbert space contractions through inner functions: a proof of Douglas's conjecture

Abstract

Let $A$ be a bounded operator on a Hilbert space and let $u$ be a non-constant Blaschke product, finite or infinite. We prove that if the spectrum of $A$ does not meet the closure of the set of poles of $u$, then $u(A)$ is similar to a contraction if and only if $A$ is. This settles a conjecture of R.G. Douglas. For infinite products $u$, no additional condition on the zeros beyond the Blaschke condition is required. We also show that '$u(A)$ is similar to a contraction if and only if $A$ is' for all non-constant inner functions $u$ whose reflected singularity set, the closure of the set of poles together with the singular support, avoids the spectrum of $A$. We also prove a weighted resolvent version of the theorem, which allows the poles to accumulate on the spectrum to a limited extent; a column criterion for partial products, which is shown to force their norm convergence; and a simultaneous version for commuting tuples of operators tested by separated inner functions sharing one Lyapunov operator. For finite Blaschke products we obtain a polynomial form of the Lyapunov operator, which gives an estimate for the similarity constants. We also characterise the tuples of finite Blaschke products that form a uniform strong test collection: they are exactly the tuples that are injective, with non-vanishing joint derivative, on the closed disc. Finally, we show by examples that the separation hypothesis cannot be replaced by the existence of a bounded $H^\infty$ calculus, by the existence of a norm radial limit, or by the strong convergence of the partial products, even for Ritt operators.