Scalability of Frames Generated by Dynamical Operators

Abstract

Let ℍ be a separable Hilbert space, let G ⊂ ℍ, and let A be an operator on ℍ. Under appropriate conditions on A and G, it is known that the set of iterations (Formula presented.) is a frame for ℍ. We call FG(A) a dynamical frame for ℍ, and explore further its properties; in particular, we show that the canonical dual frame of FG(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 FG(A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when FG(A) is a scalable frame in several special cases involving block-diagonal and companion operators.