Frames as Operator Orbits for Quaternionic Hilbert spaces

Abstract

In this paper, we study frames which can be expressed as operator orbits {T^n(φ )}n∈Zunder a single generator φ and an operator T on a right quaternionic Hilbert space H and prove a necessary and sufficient condition under which the sequence {hn}n∈Z is expressible as orbit of some operator T . Also, a necessary condition for a frame {hn}n∈Z to have an operator orbit representation
{hn}n∈Z = {T^n(h0)}n∈Z using a bounded operator T is given. Further, a characterization for the boundedness of the operator T ,
given that {hn}n∈Z = {T^n(h0)}n∈Z forms a frame is obtained. Moreover, it is proved that a redundant frame with finite excess can
never be expressed as an orbit of a bounded operator whereas for a Riesz sequence an operator orbit representation with a bounded
operator is always possible. Furthermore, we discuss the stability of frames that can be expressed as an orbit of some operator and
prove that it remains undisturbed under some perturbation conditions. Finally as an application, we approximate frames that cannot be expressed as operator orbit using the sub-orbit representation of hypercyclic operators.