Scalability of Frames Generated by Dynamical Operators

摘要

Let H be a separable Hilbert space, let G be a subset of H, and let A be an operator on H. Under appropriate conditions on A and G, it is known that the set of iterations F_G(A) is a frame for H. We call F_G(A) a dynamical frame for H, and explore further its properties; in particular, we show that its canonical dual frame 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, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. 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.