Unification, Dilation, and von Neumann’s Inequality
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Date
2025
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Department of Mathematics, NITS
Abstract
In 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.