Frames generated by the functional calculus and function frames of a normal operator

TL;DR

This paper establishes a structural equivalence between frames generated by the functional calculus of normal operators and function frames, providing new insights into their properties and applications.

Contribution

It proves that sequences generated by the functional calculus can be expressed as function sequences and characterizes their frame properties in terms of the spectrum.

Findings

Sequences can be written as function sequences $(f_n(T) g)$

Frame property characterized by the approximate point spectrum of $T^*$

Examples include operators generating Riesz bases or allowing redundancy

Abstract

In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as 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 $\sigma(T)$, 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. Our bridge theorem makes explicit the structural equivalence between frames generated by the functional calculus and function frames.