On $p$-Frames by Iterations of Operators: Duality and Perturbation

Let \(X\) be a separable Banach space with the dual space \(X^*\) and \( 1 < p < \infty\). We study some properties of \(p\)-frames for \(X\) having the form \(\{ T^n \varphi \}_{n \in I}\) where either \( I = \mathbb{N} \cup \{ 0 \}\) or \( I = \mathbb{Z}\) and \(T \colon X^* \to X^*\) is a bounded linear operator. As a main result, for a \(p\)-frame \(\{ T^k \varphi \}_{k \in \mathbb{Z}}\), we determine the structure of the canonical dual and also characterize the operator of any dual having the same construction. In the case \( I = \mathbb{N} \cup \{ 0 \}\), for a \(p\)-frame \(\{ T^n \varphi \}_{n =0}^\infty \), we get a stability result by considering perturbations of the vector \( \varphi \in X^*\) from an invariant subspace for \(T\). Finally, based on the concept of \(\varepsilon\)-approximation of a \(p\)-frame in Banach spaces, we will approximate any \(p\)-frame \(\{g_i\}_{i=1}^\infty\) via a sub-orbit of a bounded linear operator \(T\colon X^* \to X^*\) such as hypercyclic operators. This result creates a large class of \(p\)-frames generated by operator actions.