Frames generated by the functional calculus and function frames of a normal operator
Nizar El Idrissi
TL;DR
The paper studies frames generated by the functional calculus and function frames for a bounded normal operator $T$ with a cyclic vector $g$, proving a unitary equivalence to function frames on $L^2(\sigma(T),\mu_g)$ via a multiplication operator $M_z$. It then derives a spectral criterion, in terms of the approximate point spectrum of $T^*$, that exactly characterizes when such frames are automatically Riesz bases. The results unify two central constructions in operator-generated frame theory and are illustrated through normal, compact, unilateral shift, and Hardy space multiplication examples, clarifying when redundancy occurs. Overall, the work connects functional-calculus-based frames with function frames through the spectral theorem, offering a unified framework for dynamical sampling and spectral analysis.
Abstract
In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ are unitarily equivalent to function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for $T$. Here, $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors, $T$ is a bounded normal operator, $f$ and $(f_n)_{n \in \mathbb{N}}$ are functions defined on a neighborhood of the spectrum $σ(T)$, and $g$ is a cyclic vector for $T$. After that, we characterize the frame property of such sequences in terms of the approximate point spectrum of $T^*$. Examples include certain operators (normal operators, compact operators, unilateral shifts, multiplication operators on Hardy spaces, etc.) that either generate only Riesz bases or allow redundancy. Although our bridge theorem is a direct consequence of the spectral theorem for normal operators, its contribution is to make this connection explicit in the frame-theoretic context. Therefore, the aim of this paper is not to introduce new operator-theoretic techniques but to clarify and unify existing constructions.