In this paper, for a semigroup {Tt }t≥0(or group {Tt }t∈R) of operators on a Hilbert space H we investigate the properties of continuous frames of the form {Tt θ}t≥0 ({Tt θ}t∈R) for some θ ∈ H. One of the main subjects in this paper is to study the linear independence of elements of such a continuous frame. We explore conditions under which the family {Tt θ} can constitute a continuous frame when the operators in question belong to specific classes, such as normal, isometric, contraction, or self-adjoint operators. Additionally, we examine results regarding the duals of such continuous frames and provide conditions under which these duals exist. Finally, we consider various different types of perturbations of continuous frames generated by operator semigroups.

Keywords