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∞ calculus, by the existence of a norm radial limit, or by the strong convergence of the partial products, even for Ritt operators.
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BibTeX
@misc{badea2026transfer,
title = {{Transfer of similarity to Hilbert space contractions through inner functions: a proof of Douglas's conjecture}},
author = {Catalin Badea},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01703},
eprint = {2611.01703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01703},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transfer of similarity to Hilbert space contractions through inner functions: a proof of Douglas's conjecture
AU - Catalin Badea
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01703
UR - https://arxiv.org/abs/2611.01703
AB - 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^∞ calculus, by the existence of a norm radial limit, or by the strong convergence of the partial products, even for Ritt operators.
ER -