Characterizations of Classical and Quantum Weak Difference Sequence Spaces

Abstract

This paper investigates the concept of weak difference sequence spaces in normed linear spaces, focusing on both classical and quantum difference operators. We define and examine the spaces \( \mathcal{WD}(X) \) and \( \mathcal{WD}^k(X) \), which consist of sequences whose first- and higher-order differences converge weakly to zero. Additionally, we introduce a new class of sequence spaces \( \mathcal{WD}_q(X) \), governed by quantum difference operators arising from \( q \)-calculus. We establish fundamental properties of these spaces, including linearity, stability under continuous mappings and behavior under scalar multiplication. Through various examples and counterexamples, we highlight subtle distinctions between classical and quantum weak convergence. The results provide a unified framework for analyzing discrete weak convergence phenomena and lay the groundwork for further developments in sequence space theory and quantum analysis.