Scalability of frames generated by dynamical operators

TL;DR

This paper investigates the properties and scalability of dynamical frames generated by operators on Hilbert spaces, providing conditions for their duals and analyzing specific cases like normal and block-diagonal operators.

Contribution

It introduces the concept of dynamical frames, studies their duals, and characterizes conditions for their scalability, especially for normal and block-diagonal operators.

Findings

The dual of a dynamical frame also has an iterative structure.

Conditions for frame scalability depend on the operator and set G.

Specific results for normal, block-diagonal, and companion operators.

Abstract

Let $A$ be an operator on {a separable } Hilbert space $\cH$, and let $G \subset \cH$. It is known that - under appropriate conditions on $A$ and $G$ - the set of iterations $F_G(A)= \{A^j \gbf \; | \; \gbf \in G, \; 0 \leq j \leq L(\gbf) \} $ is a frame for $\cH$. We call $F_G(A)$ a dynamical frame for $\cH$, and explore further its properties; in particular, we show that the canonical dual frame of $F_G(A)$ also has an iterative set structure. We explore the relations between the operator $A$, the set $G$ and the number of iterations $L$ which ensure that the system $F_G(A)$ is a scalable frame. We give a general statement on frame scalability, We and study in detail the case when $A$ is a normal operator, utilizing the unitary diagonalization in finite dimensions. In addition, we answer the question of when $F_G(A)$ is a scalable frame in several special cases involving block-diagonal and companion operators.