A Shimorin-type analytic model on an annulus for left-invertible operators and applications

TL;DR

This work develops a Shimorin-type analytic model for left-invertible operators by realizing such operators as the multiplication by $z$ on a reproducing kernel Hilbert space of vector-valued holomorphic functions defined on an annulus (or disc). The construction hinges on a subspace $E$ with $[E]_{T^*,T'}=\mathcal{H}$ and yields a concrete unitary $U$ that intertwines $T$ with $\mathscr{M}_z$, while the Cauchy dual $T'$ corresponds to a backward-shift operator $\mathscr{L}$; convergence on annuli is characterized via radii $r^-$ and $r^+$, producing an explicit kernel $\kappa_{\mathscr{H}}$. The framework extends Shimorin's model and Gellar's bilateral shift, and it yields an improved model for weighted composition operators with finite branching index, including detailed kernel structure and radii-based convergence results. The approach provides spectral information and unifies analytic-model perspectives for left-invertible operators and a broad class of weighted composition and directed-tree shifts, with potential applications to invariant-subspace and functional-calculus questions in operator theory.

Abstract

A new analytic model for left-invertible operators, which extends both Shimorin's analytic model for left-invertible and analytic operators and Gellar's model for bilateral weighted shift is introduced and investigated. We show that a left-invertible operator $T$, which satisfies certain conditions can be modelled as a multiplication operator $\mathscr{M}_z$ on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. A similar result for composition operators in $\ell^2$-spaces is established.

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Theorems & Definitions (34)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: (Proposition 3.5.1 memo)
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3