Year: 2025
Type: article
Abstract: In this paper we shall use realization theory, a favourite technique of Rien Kaashoek, to prove new results about a class of holomorphic functions on an annulus $$ R_\delta {\mathop {=}\limits ^\textrm{def}}\{z\in \mathbb {C}: \delta<|z|<1\}, $$ R δ = def { z ∈ C : δ < | z | < 1 } , where $$0<\delta <1$$ 0 < δ < 1 . The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator T to a normal operator with spectrum in $$\partial R_\delta $$ ∂ R δ . Their work suggested the following norm $$\Vert \cdot \Vert _{\textrm{dp}}$$ ‖ · ‖ dp on the space $$\textrm{Hol}(R_\delta )$$ Hol ( R δ ) of holomorphic functions on $$R_\delta $$ R δ , $$ \Vert \varphi \Vert _{\textrm{dp}} {\mathop {=}\limits ^\textrm{def}} \sup \{ \Vert \varphi (T)\Vert : \Vert T\Vert \le 1, \Vert T^{-1}\Vert \le 1/\delta \ \text {and} \ \sigma (T)\subseteq R_\delta \}. $$ ‖ φ ‖ dp = def sup { ‖ φ ( T ) ‖ : ‖ T ‖ ≤ 1 , ‖ T - 1 ‖ ≤ 1 / δ and σ ( T ) ⊆ R δ } . By analogy with the classical Schur class of holomorphic functions $$\mathcal {S} $$ S with supremum norm at most 1 on the disc $$\mathbb {D}$$
Authors: Jim Agler, Zinaida A. Lykova, N. J. Young
Institutions: University of California San Diego, Newcastle University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4415778760