Iterative actions of normal operators

Abstract

Let A be a normal operator in a Hilbert space H, and let G ⊂ H be a countable set of vectors. We investigate the relations between A, G and L that make the system of iterations {A n g : g ∈ G, 0 ≤ n < L(g)} complete, Bessel, a basis, or a frame for H. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.

Dynamical sampling and frame representations with bounded operators

“…Example 1.1 Based on interpolation sequences in the Hardy space on the unit disc, Aldroubi et al [3] have constructed frames {T n ϕ} ∞ n=0 for ℓ 2 (N). We will formulate the result in the setting of a separable infinite-dimensional Hilbert space H. Consider an operator T of the form T = ∞ k=1 λ k P k , where P k , k ∈ N, are rank 1 orthogonal projections such that P j P k = 0, j = k, ∞ k=1 P k = I, and |λ k | < 1 for all k ∈ N. Then there exist unit vectors e k such that P k f = f, e k e k .…”

Frames induced by the action of continuous powers of an operator

“…Recent results on dynamical sampling and frames. Existing studies on various aspect of the dynamical sampling problem and related frame theory grew out of the work in [1,5,6,7,37,46], see, for example, [2,15,16,19,36,42,43,45,54,55] and the references therein. However, except for a few, they all focus on uniform discrete time-sets T ⊆ {0, 1, 2, .…”

“…On the other hand, for normal operators, it is possible to find frames of the form {A n g} g∈G, n≥0 ; however, no such a frame can be a basis [5].…”

On Regular Generalized Sampling in T-Invariant Subspaces of a Hilbert Space: An Overview

“…[14], the idea of considering frames (or Riesz) sequences of the form {T n a} n∈Z (or {T n a} N −1 n=0 ) is closely related with dynamical sampling (see, for instance, Refs. [3,4]), although the indexing of a frame in the dynamical sampling context is different from the one used here; the group structure is crucial in the sequel.…”

“…So in the finite case we have not used, explicitly, Fourier analysis. 4. In case the operator T = U is unitary, a necessary and sufficient condition for {T (h)a} h∈H to be a Riesz sequence in H can be found in [5].…”