Modifikasi Operator Weighted Backward Shift untuk Membentuk Operator Hypercyclic pada Ruang Barisan

Abstract

The linear operator T: X --> X is called an hypercyclic operator if there exist  such that the orbit of  under the operator T is dense in X. This study aims to construct an explicit hypercyclic operator on the sequence space \(\ell^p\), by modifying the backward shift operator using a non-constant weight sequence. This approach differs from Rolewicz's classical method, which used constant weights. The method applied is constructive and formal, relying on deductive reasoning in mathematical proofs. Two operators—weighted left and right shifts—are introduced and their properties are analyzed, including their composition and iterative behavior. The main result is the construction of a specific operator \(S_\bold{a}\) weighted by \(\bold{a}=(a_n)\) that is proven to be hypercyclic. The proof involves demonstrating the existence of a vector whose orbit under \(S_\bold{a}\) is dense in \(\ell^p\).