Roshan2026-11-042026-11-042025http://nits.ndl.gov.in/handle/123456789/273In this project, we present elegant proofs of four fundamental theorems using Zabreiko’s Lemma. Subsequently, we delve into Dilation Theory and von Neumann’s Inequality, exploring their diverse applications. Inspired by [1, 2], our primary objective is to explicitly check the generalizations of Ando’s dilation for a Q-commutative setup, where Q is a unitary operator. Additionally, we would like to establish a concrete connection between von Neumann’s Inequality and distinguished varieties. We aim to investigate the Q-commutative extensions of both Dilation Theory and von Neumann’s Inequality.In a Q-commutative setup, we focus on an isometric dilation of a Q−commuting pairof contractions. The present attempt is inspired by the work done by Sibaprasad Barikand Bappa Bisai on the subject ”A Generalization of Ando’s Dilation, and Isometric Dilations for a Class of Tuples of q-Commuting Contractions” [3]. We are taking aQ−commuting pair of contractions to perform Ando’s dilation modulo some conditions.enThesis